Mostly restore AIMO example to working.

lynxkite-app/data/AIMO

@@ -0,0 +1,1039 @@
+{
+  "env": "LLM logic",
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+    {
+      "id": "View 1",
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+                [
+                  "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,
+                  "Please give a correct solution for this: 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?",
+                  "To solve this problem, we need to follow these steps:\n\n1. **Understand the Parabola and Line Intersection:**\n   The parabola is given by \\( y = kx^2 - 2kx + l \\).\n   The line is given by \\( y = 4 \\).\n\n2. **Find the Intersection Points:**\n   Set the equations equal to each other to find the points of intersection \\( A \\) and \\( B \\):\n  ",
+                  2
+                ],
+                [
+                  "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,
+                  "Please give a correct solution for this: 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?",
+                  "To solve this problem, we need to ensure that any two yellow numbers have a sum that is a blue number. \n\nLet's break down the problem:\n\n1. **Understand the problem constraints:**\n   - We have three-digit numbers ranging from 111 to 999.\n   - Each number must be colored either blue or yellow.\n   - The sum of any two yellow numbers must be a blue number.\n\n2. **Define the variables:**\n   -",
+                  2
+                ],
+                [
+                  "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,
+                  "Please give a correct solution for this: 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?",
+                  "Let's break down the problem step by step. \n\n**Understanding the Sparkle Operation:**\n\n1. **Sum of Digits:**\n   For a number \\( n \\), the sum of its digits is calculated.\n\n2. **Factorial of Sum:**\n   The sparkle is then calculated as \\( n! \\).\n\n**Special Numbers:**\n\nA special number is defined as one that eventually leads to every subsequent number being less than 6. This implies",
+                  2
+                ],
+                [
+                  "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,
+                  "Please give a correct solution for this: What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?",
+                  "To solve the problem, we need to find the minimum value of the expression \\(5x^2 + 5y^2 - 8xy\\) subject to the constraint \\(|x - 2y| + |y - 2x| = 40\\).\n\nHere are the steps to solve this problem:\n\n1. **Understand the Constraint:**\n   The constraint given is \\(|x - 2y| + |y - 2",
+                  2
+                ],
+                [
+                  "5277ed",
+                  "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?",
+                  211,
+                  "Please give a correct solution for this: There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?",
+                  "To solve this problem, we need to find a unique increasing geometric sequence of five 2-digit positive integers. Let's denote the first term of the sequence by \\( a \\) and the common ratio by \\( r \\). The five terms of the sequence can be represented as:\n\n\\[ a, ar, ar^2, ar^3, ar^4 \\]\n\nGiven that all these terms are 2-digit positive integers, we need to find \\",
+                  2
+                ],
+                [
+                  "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,
+                  "Please give a correct solution for this: For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?",
+                  "To solve the problem, we need to analyze the equation \\(\\vert \\vert x-1 \\vert - 2 \\vert = \\frac{m}{100}\\) and determine for how many positive integers \\(m\\) the equation has exactly 4 distinct solutions.\n\n### Steps to Solve the Problem:\n\n1. **Understand the Absolute Value Equation:**\n   - The equation \\(\\vert \\vert x-1 \\vert - 2 \\vert = \\frac",
+                  2
+                ],
+                [
+                  "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,
+                  "Please give a correct solution for this: 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 $a + b$.",
+                  "To solve this problem, we need to determine the probability \\(\\frac{a}{b}\\) that the highest roll on four 6-sided dice is a 5, and then find the values of \\(a\\) and \\(b\\) such that \\(a\\) and \\(b\\) are relatively prime. Let's break down the steps:\n\n1. **Determine the Sample Space:**\n   - Calculate the total number of possible outcomes when rolling four ",
+                  2
+                ],
+                [
+                  "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,
+                  "Please give a correct solution for this: 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?",
+                  "To solve this problem, we need to find the points \\((x, y)\\) that satisfy the given equation:\n\\[\n((\\text{Abs}(x + y) - 10)^2 + (\\text{Abs}(x - y) - 10)^2)((\\text{Abs}(x) - 8)^2 + (\\text{Abs}(y) - 8)^2) = 0\n\\]\n\nFirst,",
+                  2
+                ],
+                [
+                  "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,
+                  "Please give a correct solution for this: 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.",
+                  "To solve this problem, we need to carefully analyze the geometric configuration and calculate the areas of the four regions formed by the lines \\(DP\\) and \\(BQ\\) within the unit square \\(ABCD\\).\n\nLet's break down the problem into manageable parts:\n\n1. **Define the Coordinates:**\n   - Place the unit square \\(ABCD\\) in the coordinate plane with \\(A\\) at \\((0, 0)\\),",
+                  2
+                ],
+                [
+                  "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,
+                  "Please give a correct solution for this: 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 $f(100)$.",
+                  "To solve the problem, we need to find the function \\( f \\) that satisfies the given conditions:\n\n1. \\( f(f(f(n))) = 8n - 7 \\)\n2. \\( f(2n) = 2f(n) + 1 \\)\n\nLet's break down the problem step-by-step:\n\n### Step 1: Analyze the Recursion\n\nFirst, let's express",
+                  2
+                ]
+              ]
+            }
+          }
+        },
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+                [
+                  "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?",
+                  52.0,
+                  "Please give a correct solution for this: 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?",
+                  " We start by finding the x-coordinates of points"
+                ],
+                [
+                  "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?",
+                  250.0,
+                  "Please give a correct solution for this: 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?",
+                  " Let's consider the numbers in the form of"
+                ],
+                [
+                  "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?",
+                  702.0,
+                  "Please give a correct solution for this: 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?",
+                  " Let $S$ denote the set of all $"
+                ],
+                [
+                  "430b63",
+                  "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?",
+                  800.0,
+                  "Please give a correct solution for this: What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?",
+                  " We can rewrite the given equation as $|x-"
+                ],
+                [
+                  "5277ed",
+                  "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?",
+                  211.0,
+                  "Please give a correct solution for this: There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?",
+                  " Let the five terms of the geometric sequence be $"
+                ],
+                [
+                  "739bc9",
+                  "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?",
+                  199.0,
+                  "Please give a correct solution for this: For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?",
+                  " Let's break down the problem.\n\n"
+                ],
+                [
+                  "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 $a + b$.",
+                  185.0,
+                  "Please give a correct solution for this: 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 $a + b$.",
+                  " The total number of outcomes when rolling four 6"
+                ],
+                [
+                  "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?",
+                  320.0,
+                  "Please give a correct solution for this: 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?",
+                  " We see that the given equation is equivalent to either"
+                ],
+                [
+                  "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.",
+                  480.0,
+                  "Please give a correct solution for this: 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.",
+                  " [asy] size(7cm); pair A"
+                ],
+                [
+                  "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 $f(100)$.",
+                  199.0,
+                  "Please give a correct solution for this: 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 $f(100)$.",
+                  " Let $P(n)$ be the assertion that"
+                ]
+              ]
+            }
+          }
+        }
+      },
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+                  "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. How many such special numbers are there with at most 36 digits?",
+                  702
+                ],
+                [
+                  "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
+                ],
+                [
+                  "5277ed",
+                  "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?",
+                  211
+                ],
+                [
+                  "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
+                ],
+                [
+                  "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
+                ],
+                [
+                  "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
+                ],
+                [
+                  "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
+                ],
+                [
+                  "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
+                ]
+              ]
+            }
+          }
+        },
+        "error": null,
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+                  "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?",
+                  52.0
+                ],
+                [
+                  "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?",
+                  250.0
+                ],
+                [
+                  "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?",
+                  702.0
+                ],
+                [
+                  "430b63",
+                  "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?",
+                  800.0
+                ],
+                [
+                  "5277ed",
+                  "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?",
+                  211.0
+                ],
+                [
+                  "739bc9",
+                  "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?",
+                  199.0
+                ],
+                [
+                  "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 $a + b$.",
+                  185.0
+                ],
+                [
+                  "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?",
+                  320.0
+                ],
+                [
+                  "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.",
+                  480.0
+                ],
+                [
+                  "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 $f(100)$.",
+                  199.0
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+    },
+    {
+      "id": "Create prompt 2",
+      "type": "basic",
+      "data": {
+        "title": "Create prompt",
+        "params": {
+          "template": "Is this a nice solution? {{response}}"
+        },
+        "display": null,
+        "error": null,
+        "__execution_delay": 0.0,
+        "meta": {
+          "inputs": {
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+                "type": "None"
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+    {
+      "id": "View 3",
+      "type": "table_view",
+      "data": {
+        "title": "View",
+        "params": {},
+        "display": {
+          "dataframes": {
+            "df": {
+              "columns": [
+                "id",
+                "text",
+                "answer",
+                "prompt",
+                "response",
+                "iterations-Loop 1"
+              ],
+              "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 this problem, we need to follow these steps:\n\n1. **Understand the Parabola and Line Intersection:**\n   The parabola is given by \\( y = kx^2 - 2kx + l \\).\n   The line is given by \\( y = 4 \\).\n\n2. **Find the Intersection Points:**\n   Set the equations equal to each other to find the points of intersection \\( A \\) and \\( B \\):\n  ",
+                  "To solve this problem, we need to set the equation of the parabola equal to the equation of the line and solve for the points of intersection. Let's break it down into the steps you've outlined and use Python with SymPy to find the solution.\n\n### Steps:\n\n1. **Understand the Parabola and Line Intersection:**\n   - The equation of the parabola is \\( y = kx^2 - 2kx + l \\).\n   - The",
+                  3
+                ],
+                [
+                  "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 this problem, we need to ensure that any two yellow numbers have a sum that is a blue number. \n\nLet's break down the problem:\n\n1. **Understand the problem constraints:**\n   - We have three-digit numbers ranging from 111 to 999.\n   - Each number must be colored either blue or yellow.\n   - The sum of any two yellow numbers must be a blue number.\n\n2. **Define the variables:**\n   -",
+                  "To solve this problem, we need to ensure that for any two yellow numbers, their sum falls within the range of blue numbers. Let's break down the problem step-by-step.\n\n### Problem Breakdown\n\n1. **Define the Variables:**\n   - Let \\( Y \\) be a set of three-digit yellow numbers.\n   - Let \\( B \\) be a set of three-digit blue numbers.\n   - Let \\( \\text{Sum}(a",
+                  3
+                ],
+                [
+                  "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? Let's break down the problem step by step. \n\n**Understanding the Sparkle Operation:**\n\n1. **Sum of Digits:**\n   For a number \\( n \\), the sum of its digits is calculated.\n\n2. **Factorial of Sum:**\n   The sparkle is then calculated as \\( n! \\).\n\n**Special Numbers:**\n\nA special number is defined as one that eventually leads to every subsequent number being less than 6. This implies",
+                  "The solution you've provided for calculating a Sparkle number is a structured approach to solving the problem, but it could be improved in several ways to ensure clarity, efficiency, and completeness. Let's break down the steps and identify potential improvements:\n\n1. **Understanding the Problem:**\n   - **Sum of Digits:** Calculate the sum of the digits of a number \\( n \\).\n   - **Factorial Calculation:** Compute the factorial of the sum of digits.\n\n2",
+                  3
+                ],
+                [
+                  "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 problem, we need to find the minimum value of the expression \\(5x^2 + 5y^2 - 8xy\\) subject to the constraint \\(|x - 2y| + |y - 2x| = 40\\).\n\nHere are the steps to solve this problem:\n\n1. **Understand the Constraint:**\n   The constraint given is \\(|x - 2y| + |y - 2",
+                  "To solve this optimization problem with the given constraint, we can use the method of Lagrange multipliers. This technique is useful when we have a function to maximize or minimize subject to equality constraints.\n\nGiven:\n\\[ f(x, y) = 5x^2 + 5y^2 - 8xy \\]\n\\[ g(x, y) = |x - 2y| + |y - 2x| = 40 \\",
+                  3
+                ],
+                [
+                  "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 this problem, we need to find a unique increasing geometric sequence of five 2-digit positive integers. Let's denote the first term of the sequence by \\( a \\) and the common ratio by \\( r \\). The five terms of the sequence can be represented as:\n\n\\[ a, ar, ar^2, ar^3, ar^4 \\]\n\nGiven that all these terms are 2-digit positive integers, we need to find \\",
+                  "To solve the problem of finding a unique increasing geometric sequence of five 2-digit positive integers \\(a, ar, ar^2, ar^3, ar^4\\), we can follow these steps:\n\n1. **Understand the Problem:**\n   We need to find \\(a\\) and \\(r\\) such that \\(a, ar, ar^2, ar^3, ar^4\\) are all 2-digit positive integers. Let",
+                  3
+                ],
+                [
+                  "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 problem, we need to analyze the equation \\(\\vert \\vert x-1 \\vert - 2 \\vert = \\frac{m}{100}\\) and determine for how many positive integers \\(m\\) the equation has exactly 4 distinct solutions.\n\n### Steps to Solve the Problem:\n\n1. **Understand the Absolute Value Equation:**\n   - The equation \\(\\vert \\vert x-1 \\vert - 2 \\vert = \\frac",
+                  "To solve the problem, we need to determine the number of positive integers \\( m \\) for which the equation \\(\\vert \\vert x-1 \\vert - 2 \\vert = \\frac{m}{100}\\) has exactly 4 distinct solutions for \\( x \\).\n\n### Step-by-Step Solution:\n\n1. **Analyze the Absolute Value Equation:**\n   The equation \\(\\vert \\vert x-1 \\vert - 2 \\",
+                  3
+                ],
+                [
+                  "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 this problem, we need to determine the probability \\(\\frac{a}{b}\\) that the highest roll on four 6-sided dice is a 5, and then find the values of \\(a\\) and \\(b\\) such that \\(a\\) and \\(b\\) are relatively prime. Let's break down the steps:\n\n1. **Determine the Sample Space:**\n   - Calculate the total number of possible outcomes when rolling four ",
+                  "Sure, to determine if the solution provided is \"nice\" in the context of being clear, concise, and well-organized, let's evaluate the solution for clarity and completeness:\n\n### Solution Breakdown\n\n1. **Determine the Sample Space:**\n   - **Calculate the Total Number of Possible Outcomes:**\n     Rolling four 6-sided dice means each die has 6 possible outcomes. Therefore, the total number of possible outcomes is:\n     \\[\n     6 \\times ",
+                  3
+                ],
+                [
+                  "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 this problem, we need to find the points \\((x, y)\\) that satisfy the given equation:\n\\[\n((\\text{Abs}(x + y) - 10)^2 + (\\text{Abs}(x - y) - 10)^2)((\\text{Abs}(x) - 8)^2 + (\\text{Abs}(y) - 8)^2) = 0\n\\]\n\nFirst,",
+                  "To determine if the given solution is \"nice\" (i.e., provides integer or simple coordinates), we need to analyze the equation:\n\n\\[\n((\\text{Abs}(x + y) - 10)^2 + (\\text{Abs}(x - y) - 10)^2)((\\text{Abs}(x) - 8)^2 + (\\text{Abs}(y) - 8)^2) = 0\n",
+                  3
+                ],
+                [
+                  "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 this problem, we need to carefully analyze the geometric configuration and calculate the areas of the four regions formed by the lines \\(DP\\) and \\(BQ\\) within the unit square \\(ABCD\\).\n\nLet's break down the problem into manageable parts:\n\n1. **Define the Coordinates:**\n   - Place the unit square \\(ABCD\\) in the coordinate plane with \\(A\\) at \\((0, 0)\\),",
+                  "Yes, this is a nice and clear solution to the problem of calculating the areas of the four regions formed by the lines \\(DP\\) and \\(BQ\\) within a unit square \\(ABCD\\). Here's a step-by-step breakdown of the approach:\n\n1. **Define the Coordinates:**\n   - Place the unit square \\(ABCD\\) in the coordinate plane with \\(A\\) at \\((0, 0)\\),",
+                  3
+                ],
+                [
+                  "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 problem, we need to find the function \\( f \\) that satisfies the given conditions:\n\n1. \\( f(f(f(n))) = 8n - 7 \\)\n2. \\( f(2n) = 2f(n) + 1 \\)\n\nLet's break down the problem step-by-step:\n\n### Step 1: Analyze the Recursion\n\nFirst, let's express",
+                  "To solve the problem, we need to find a function \\( f \\) that satisfies the given conditions:\n\n1. \\( f(f(f(n))) = 8n - 7 \\)\n2. \\( f(2n) = 2f(n) + 1 \\)\n\nLet's break down the problem step-by-step:\n\n### Step 1: Analyze the Recursion\n\nGiven the recursive nature of",
+                  3
+                ]
+              ]
+            }
+          }
+        },
+        "error": null,
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+                  "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?",
+                  52.0,
+                  "Is this a nice solution?  We start by finding the x-coordinates of points",
+                  "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?",
+                  250.0,
+                  "Is this a nice solution?  Let's consider the numbers in the form of",
+                  "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?",
+                  702.0,
+                  "Is this a nice solution?  Let $S$ denote the set of all $",
+                  "no"
+                ],
+                [
+                  "430b63",
+                  "What is the minimum value of $5x^2+5y^2-8xy$ when $x$ and $y$ range over all real numbers such that $|x-2y| + |y-2x| = 40$?",
+                  800.0,
+                  "Is this a nice solution?  We can rewrite the given equation as $|x-",
+                  "no"
+                ],
+                [
+                  "5277ed",
+                  "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?",
+                  211.0,
+                  "Is this a nice solution?  Let the five terms of the geometric sequence be $",
+                  "yes"
+                ],
+                [
+                  "739bc9",
+                  "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?",
+                  199.0,
+                  "Is this a nice solution?  Let's break down the problem.\n\n",
+                  "yes"
+                ],
+                [
+                  "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 $a + b$.",
+                  185.0,
+                  "Is this a nice solution?  The total number of outcomes when rolling four 6",
+                  "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?",
+                  320.0,
+                  "Is this a nice solution?  We see that the given equation is equivalent to either",
+                  "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.",
+                  480.0,
+                  "Is this a nice solution?  [asy] size(7cm); pair A",
+                  "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 $f(100)$.",
+                  199.0,
+                  "Is this a nice solution?  Let $P(n)$ be the assertion that",
+                  "yes"
+                ]
+              ]
+            }
+          }
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+          "name": "View",
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+      "type": "basic",
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+        "title": "Input CSV",
+        "params": {
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+          "key": "problem"
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+      "id": "Ask LLM 1",
+      "type": "basic",
+      "data": {
+        "title": "Ask LLM",
+        "params": {
+          "model": "SultanR/SmolTulu-1.7b-Instruct",
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+              "position": "right",
+              "name": "output",
+              "type": {
+                "type": "None"
+              }
+            }
+          },
+          "name": "Ask LLM",
+          "inputs": {
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+      "id": "Input CSV 1 View 2",
+      "source": "Input CSV 1",
+      "target": "View 2",
+      "sourceHandle": "output",
+      "targetHandle": "input"
+    },
+    {
+      "id": "Input CSV 1 Create prompt 1",
+      "source": "Input CSV 1",
+      "target": "Create prompt 1",
+      "sourceHandle": "output",
+      "targetHandle": "input"
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+    {
+      "id": "Create prompt 1 Ask LLM 3",
+      "source": "Create prompt 1",
+      "target": "Ask LLM 3",
+      "sourceHandle": "output",
+      "targetHandle": "input"
+    },
+    {
+      "id": "Ask LLM 3 Create prompt 2",
+      "source": "Ask LLM 3",
+      "target": "Create prompt 2",
+      "sourceHandle": "output",
+      "targetHandle": "input"
+    },
+    {
+      "id": "Ask LLM 3 Loop 1",
+      "source": "Ask LLM 3",
+      "target": "Loop 1",
+      "sourceHandle": "output",
+      "targetHandle": "input"
+    },
+    {
+      "id": "Ask LLM 3 View 1",
+      "source": "Ask LLM 3",
+      "target": "View 1",
+      "sourceHandle": "output",
+      "targetHandle": "input"
+    },
+    {
+      "id": "Create prompt 2 Ask LLM 1",
+      "source": "Create prompt 2",
+      "target": "Ask LLM 1",
+      "sourceHandle": "output",
+      "targetHandle": "input"
+    },
+    {
+      "id": "Ask LLM 1 View 3",
+      "source": "Ask LLM 1",
+      "target": "View 3",
+      "sourceHandle": "output",
+      "targetHandle": "input"
+    }
+  ]
+}

lynxkite-app/data/aimo-examples.csv

@@ -0,0 +1,11 @@
+"id","problem","answer"
+"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. How many such special numbers are there with at most 36 digits?",702
+"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
+"5277ed","There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?",211
+"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
+"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
+"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
+"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
+"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

lynxkite-lynxscribe/src/lynxkite_plugins/lynxscribe/llm_ops.py

@@ -120,7 +120,7 @@ def ask_llm(input, *, model: str, accepted_regex: str = None, max_tokens: int =
     options = {}
     if accepted_regex:
         options["extra_body"] = {
-            "guided_regex": accepted_regex,
+            "regex": accepted_regex,
         }
     results = chat(
         model=model,