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| "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 | |