diff --git "a/data/Graph RAG" "b/data/Graph RAG" new file mode 100644--- /dev/null +++ "b/data/Graph RAG" @@ -0,0 +1,2917 @@ +{ + "env": "LLM logic", + "nodes": [ + { + "id": "Input 1", + "type": "basic", + "data": { + "title": "Input", + "params": { + "filename": "/Users/danieldarabos/Downloads/aimo-train.csv", + "key": "problem" + }, + "display": null, + "error": null, + "meta": { + "name": "Input", + "params": { + "filename": { + "name": "filename", + "default": null, + "type": { + "format": "path" + } + }, + "key": { + "name": "key", + "default": null, + "type": { + "type": "" + } + } + }, + "inputs": {}, + "outputs": { + "output": { + "name": "output", + "type": { + "type": "None" + }, + "position": "right" + } + }, + "type": "basic", + "sub_nodes": null + } + }, + "position": { + "x": -663.9814415296167, + "y": -24.673405923964822 + }, + "parentId": null, + "width": 259, + "height": 478 + }, + { + "id": "Create prompt 1", + "type": "basic", + "data": { + "title": "Create prompt", + "params": { + "save_as": "prompt", + "template": "Similar problems:\n\n {% for item in rag %}\n{{ item.text }}\n{% endfor %}\n\nWhat would be the first step of solving the following problem? {{text}}" + }, + "display": null, + "error": null, + "meta": { + "name": "Create prompt", + "params": { + "save_as": { + "name": "save_as", + "default": "prompt", + "type": { + "type": "" + } + }, + "template": { + "name": "template", + "default": null, + "type": { + "format": "textarea" + } + } + }, + "inputs": { + "input": { + "name": "input", + "type": { + "type": "" + }, + "position": "left" + } + }, + "outputs": { + "output": { + "name": "output", + "type": { + "type": "None" + }, + "position": "right" + } + }, + "type": "basic", + "sub_nodes": null + }, + "beingResized": false, + "collapsed": false + }, + "position": { + "x": 92.48251817426697, + "y": 2.986072034314958 + }, + "parentId": null, + "width": 300, + "height": 328 + }, + { + "id": "Create prompt 2", + "type": "basic", + "data": { + "title": "Create prompt", + "params": { + "save_as": "prompt", + "template": "Is this a nice solution? {{response}}" + }, + "display": null, + "error": null, + "meta": { + "name": "Create prompt", + "params": { + "save_as": { + "name": "save_as", + "default": "prompt", + "type": { + "type": "" + } + }, + "template": { + "name": "template", + "default": null, + "type": { + "format": "textarea" + } + } + }, + "inputs": { + "input": { + "name": "input", + "type": { + "type": "" + }, + "position": "left" + } + }, + "outputs": { + "output": { + "name": "output", + "type": { + "type": "None" + }, + "position": "right" + } + }, + "type": "basic", + "sub_nodes": null + } + }, + "position": { + "x": 863.0737654843731, + "y": -68.00791743959714 + }, + "parentId": null + }, + { + "id": "View 3", + "type": "table_view", + "data": { + "title": "View", + "params": {}, + "display": { + "dataframes": { + "df": { + "columns": [ + "id", + "text", + "answer", + "rag", + "prompt", + "response" + ], + "data": [ + [ + "229ee8", + "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + 52, + [ + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + } + ], + "Is this a nice solution? 0.7139497827292313", + "0.19053007335807304" + ], + [ + "246d26", + "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + 250, + [ + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + } + ], + "Is this a nice solution? 0.07328817086993833", + "0.7216846012621502" + ], + [ + "2fc4ad", + "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + 702, + [ + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + } + ], + "Is this a nice solution? 0.032316140702087215", + "0.17170311684440376" + ], + [ + "430b63", + "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + 800, + [ + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + } + ], + "Is this a nice solution? 0.31766897160862206", + "0.19256153092019734" + ], + [ + "5277ed", + "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + 211, + [ + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + } + ], + "Is this a nice solution? 0.9811958281016008", + "0.8688809804290338" + ], + [ + "739bc9", + "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + 199, + [ + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + } + ], + "Is this a nice solution? 0.38108403438574845", + "0.09979627018489712" + ], + [ + "82e2a0", + "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + 185, + [ + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + } + ], + "Is this a nice solution? 0.42948749188004376", + "0.772154996120189" + ], + [ + "8ee6f3", + "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + 320, + [ + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + } + ], + "Is this a nice solution? 0.3487100303819337", + "0.5386589252803151" + ], + [ + "bedda4", + "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + 480, + [ + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + } + ], + "Is this a nice solution? 0.192928132428547", + "0.5589770560628611" + ], + [ + "d7e9c9", + "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + 199, + [ + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + } + ], + "Is this a nice solution? 0.20024247107489412", + "0.3107009777355463" + ], + [ + "246d26", + "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + 250, + [ + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + } + ], + "Is this a nice solution? 0.07328817086993833", + "0.7216846012621502" + ], + [ + "2fc4ad", + "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + 702, + [ + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + } + ], + "Is this a nice solution? 0.032316140702087215", + "0.17170311684440376" + ], + [ + "430b63", + "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + 800, + [ + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + } + ], + "Is this a nice solution? 0.31766897160862206", + "0.19256153092019734" + ], + [ + "5277ed", + "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + 211, + [ + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + } + ], + "Is this a nice solution? 0.9811958281016008", + "0.8688809804290338" + ], + [ + "739bc9", + "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + 199, + [ + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + } + ], + "Is this a nice solution? 0.38108403438574845", + "0.09979627018489712" + ], + [ + "82e2a0", + "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + 185, + [ + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + } + ], + "Is this a nice solution? 0.42948749188004376", + "0.772154996120189" + ], + [ + "8ee6f3", + "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + 320, + [ + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + } + ], + "Is this a nice solution? 0.3487100303819337", + "0.5386589252803151" + ], + [ + "bedda4", + "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + 480, + [ + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + } + ], + "Is this a nice solution? 0.192928132428547", + "0.5589770560628611" + ], + [ + "d7e9c9", + "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + 199, + [ + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + } + ], + "Is this a nice solution? 0.20024247107489412", + "0.3107009777355463" + ] + ] + } + } + }, + "error": null, + "meta": { + "name": "View", + "params": {}, + "inputs": { + "input": { + "name": "input", + "type": { + "type": "" + }, + "position": "left" + } + }, + "outputs": {}, + "type": "table_view", + "sub_nodes": null + }, + "view": { + "dataframes": { + "df": { + "columns": [ + "id", + "text", + "answer", + "prompt", + "response" + ], + "data": [ + [ + "229ee8", + "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + 52, + "Is this a nice solution? To solve the given problem, the first step would", + "no" + ], + [ + "246d26", + "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + 250, + "Is this a nice solution? To solve the given problem, the first step would", + "no" + ], + [ + "2fc4ad", + "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + 702, + "Is this a nice solution? To solve the given problem, the first step would", + "no" + ], + [ + "430b63", + "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + 800, + "Is this a nice solution? To solve the given problem, the first step would", + "no" + ], + [ + "5277ed", + "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + 211, + "Is this a nice solution? To solve the given problem, the first step would", + "no" + ], + [ + "739bc9", + "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + 199, + "Is this a nice solution? To solve the given problem, the first step would", + "no" + ], + [ + "82e2a0", + "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + 185, + "Is this a nice solution? To solve the given problem, the first step would", + "no" + ], + [ + "8ee6f3", + "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + 320, + "Is this a nice solution? To solve the given problem, the first step would", + "no" + ], + [ + "bedda4", + "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + 480, + "Is this a nice solution? To solve the given problem, the first step would", + "no" + ], + [ + "d7e9c9", + "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + 199, + "Is this a nice solution? To solve the given problem, the first step would", + "no" + ] + ] + } + } + }, + "beingResized": false + }, + "position": { + "x": 1611.9087959003862, + "y": 73.5764910384212 + }, + "parentId": null, + "width": 659, + "height": 346 + }, + { + "id": "View 2", + "type": "table_view", + "data": { + "title": "View", + "params": {}, + "display": { + "dataframes": { + "df": { + "columns": [ + "id", + "text", + "answer", + "rag", + "prompt" + ], + "data": [ + [ + "229ee8", + "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + 52, + [ + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + } + ], + "Similar problems:\n\n \nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?\n\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?\n\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.\n\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?\n\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?\n\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?\n\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.\n\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?\n\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?\n\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.\n\n\nWhat would be the first step of solving the following problem? Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?" + ], + [ + "246d26", + "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + 250, + [ + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + } + ], + "Similar problems:\n\n \nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?\n\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?\n\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?\n\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?\n\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.\n\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?\n\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.\n\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?\n\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?\n\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.\n\n\nWhat would be the first step of solving the following problem? Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?" + ], + [ + "2fc4ad", + "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + 702, + [ + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + } + ], + "Similar problems:\n\n \nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?\n\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?\n\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?\n\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.\n\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.\n\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?\n\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?\n\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?\n\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?\n\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.\n\n\nWhat would be the first step of solving the following problem? Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?" + ], + [ + "430b63", + "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + 800, + [ + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + } + ], + "Similar problems:\n\n \nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?\n\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?\n\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.\n\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?\n\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?\n\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.\n\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?\n\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.\n\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?\n\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?\n\n\nWhat would be the first step of solving the following problem? What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?" + ], + [ + "5277ed", + "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + 211, + [ + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + } + ], + "Similar problems:\n\n \nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?\n\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?\n\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?\n\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.\n\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.\n\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?\n\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?\n\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?\n\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?\n\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.\n\n\nWhat would be the first step of solving the following problem? There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?" + ], + [ + "739bc9", + "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + 199, + [ + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + } + ], + "Similar problems:\n\n \nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?\n\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.\n\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?\n\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?\n\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?\n\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?\n\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?\n\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?\n\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.\n\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.\n\n\nWhat would be the first step of solving the following problem? For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?" + ], + [ + "82e2a0", + "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + 185, + [ + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + } + ], + "Similar problems:\n\n \nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.\n\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?\n\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?\n\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.\n\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?\n\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?\n\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?\n\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?\n\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?\n\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.\n\n\nWhat would be the first step of solving the following problem? Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$." + ], + [ + "8ee6f3", + "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + 320, + [ + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + } + ], + "Similar problems:\n\n \nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?\n\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.\n\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?\n\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?\n\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?\n\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?\n\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?\n\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.\n\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.\n\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?\n\n\nWhat would be the first step of solving the following problem? The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?" + ], + [ + "bedda4", + "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + 480, + [ + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + } + ], + "Similar problems:\n\n \nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.\n\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?\n\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?\n\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?\n\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.\n\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?\n\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?\n\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?\n\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?\n\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.\n\n\nWhat would be the first step of solving the following problem? Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region." + ], + [ + "d7e9c9", + "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + 199, + [ + { + "id": "d7e9c9", + "text": "A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.." + }, + { + "id": "739bc9", + "text": "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?", + "answer": 199, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?." + }, + { + "id": "2fc4ad", + "text": "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?", + "answer": 702, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?." + }, + { + "id": "5277ed", + "text": "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?", + "answer": 211, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?." + }, + { + "id": "246d26", + "text": "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + "answer": 250, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?." + }, + { + "id": "430b63", + "text": "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?", + "answer": 800, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?." + }, + { + "id": "8ee6f3", + "text": "The points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?", + "answer": 320, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?." + }, + { + "id": "229ee8", + "text": "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + "answer": 52, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?." + }, + { + "id": "82e2a0", + "text": "Suppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", + "answer": 185, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.." + }, + { + "id": "bedda4", + "text": "Let $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.", + "answer": 480, + "prompt": "Here's some examples:\n...\nCreate a similar solution for the following problem:\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.." + } + ], + "Similar problems:\n\n \nA function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$.\n\nFor how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?\n\nLet the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. How many such special numbers are there with at most 36 digits?\n\nThere exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?\n\nEach of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?\n\nWhat is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?\n\nThe points $\\left(x, y\\right)$ satisfying $((\\vert x + y \\vert - 10)^2 + ( \\vert x - y \\vert - 10)^2)((\\vert x \\vert - 8)^2 + ( \\vert y \\vert - 8)^2) = 0$ enclose a convex polygon. What is the area of this convex polygon?\n\nLet $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?\n\nSuppose that we roll four 6-sided fair dice with faces numbered 1 to~6. Let $a/b$ be the probability that the highest roll is a 5, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.\n\nLet $ABCD$ be a unit square. Let $P$ be the point on $AB$ such that $|AP| = 1/{20}$ and let $Q$ be the point on $AD$ such that $|AQ| = 1/{24}$. The lines $DP$ and $BQ$ divide the square into four regions. Find the ratio between the areas of the largest region and the smallest region.\n\n\nWhat would be the first step of solving the following problem? A function $f: \\mathbb N \\to \\mathbb N$ satisfies the following two conditions for all positive integers $n$:$f(f(f(n)))=8n-7$ and $f(2n)=2f(n)+1$. Calculate $f(100)$." + ] + ] + } + } + }, + "error": null, + "meta": { + "name": "View", + "params": {}, + "inputs": { + "input": { + "name": "input", + "type": { + "type": "" + }, + "position": "left" + } + }, + "outputs": {}, + "type": "table_view", + "sub_nodes": null + }, + "view": { + "dataframes": { + "df": { + "columns": [ + "id", + "text", + "answer" + ], + "data": [ + [ + "229ee8", + "Let $k, l > 0$ be parameters. The parabola $y = kx^2 - 2kx + l$ intersects the line $y = 4$ at two points $A$ and $B$. These points are distance 6 apart. What is the sum of the squares of the distances from $A$ and $B$ to the origin?", + 52 + ], + [ + "246d26", + "Each of the three-digits numbers $111$ to $999$ is coloured blue or yellow in such a way that the sum of any two (not necessarily different) yellow numbers is equal to a blue number. What is the maximum possible number of yellow numbers there can be?", + 250 + ], + [ + "2fc4ad", + "Let the `sparkle' operation on positive integer $n$ consist of calculating the sum of the digits of $n$ and taking its factorial, e.g. the sparkle of 13 is $4! = 24$. A robot starts with a positive integer on a blackboard, then after each second for the rest of eternity, replaces the number on the board with its sparkle. For some `special' numbers, if they're the first number, then eventually every number that appears will be less than 6. 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