examples/AIMO.lynxkite.json

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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 find the points of intersection \\( A \\) and \\( B \\) of the parabola \\( y = kx^2 - 2kx + l \\) and the line \\( y = 4 \\), we set the equations equal:\n\n\\[\nkx^2 - 2kx + l = 4\n\\]\n\nRearranging this equation, we have:\n\n\\[\nkx^2 - 2kx + (l - 4) = 0\n\\]\n\nThis is"
                ],
                [
                  "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 start by letting \\( Y \\) denote the set of yellow numbers and \\( B \\) denote the set of blue numbers. The condition provided states that for any two yellow numbers \\( y_1, y_2 \\in Y \\), their sum \\( y_1 + y_2 \\) must equal some blue number \\( b \\in B \\). \n\nGiven that we are working with three-digit numbers, we note that any number \\( y \\) in \\("
                ],
                [
                  "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?",
                  "To solve the problem, we first need to understand the sparkle operation performed on a positive integer \\( n \\). The sparkle of \\( n \\), denoted as \\( S(n) \\), is defined as the sum of its digits, denoted \\( s(n) \\), followed by taking the factorial of that sum, i.e., \n\n\\[\nS(n) = s(n)!\n\\]\n\nWe are tasked with finding \"special\" numbers \u2014 those numbers such that starting from them, all subsequent values"
                ],
                [
                  "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 $|x-2y| + |y-2x| = 40$?",
                  "We need to minimize the expression \\( f(x, y) = 5x^2 + 5y^2 - 8xy \\) under the constraint given by the equation \\( |x - 2y| + |y - 2x| = 40 \\).\n\nFirst, we analyze the constraint \\( |x - 2y| + |y - 2x| = 40 \\). This absolute value constraint suggests there are multiple cases to consider. Let's define the"
                ],
                [
                  "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 find a unique increasing geometric sequence of five 2-digit integers, we can denote the integers in the sequence as follows:\n\n\\[\na, ar, ar^2, ar^3, ar^4\n\\]\n\nwhere \\(a\\) is the first term and \\(r\\) is the common ratio.\n\nSince we want each term to be a two-digit integer, we have the following constraints:\n\n\\[\n10 \\leq a < 100\n\\]\n\\[\n10 \\le"
                ],
                [
                  "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 equation \\( \\vert \\vert x-1 \\vert -2 \\vert = \\frac{m}{100} \\) for the number of positive integers \\( m \\) that yield \\( 4 \\) distinct solutions, we first analyze the expression \\( \\vert \\vert x-1 \\vert -2 \\vert \\).\n\n1. **Break down the expression:**\n   Let \\( y = \\vert x - 1 \\vert \\). Then, we rewrite the equation as"
                ],
                [
                  "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 $a + b$.",
                  "Given that we want the probability that the highest roll of four 6-sided dice is exactly 5, we need to consider two things: that at least one die shows a 5 and no die shows a 6.\n\n### Step 1: Calculate the total number of outcomes when rolling four dice.\nEach die has 6 faces, so when we roll four dice, the total number of possible outcomes is:\n\n\\[\n6^4 = 1296.\n\\]\n\n### Step 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 the equation \n\n\\[\n((|x + y| - 10)^2 + (|x - y| - 10)^2)((|x| - 8)^2 + (|y| - 8)^2) = 0,\n\\]\n\nwe need to analyze the two factors separately and determine the resultant sets of points.\n\n### Step 1: Analyze the first factor\n\nThe expression \\((|x + y| - 10)^2 + (|x"
                ],
                [
                  "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 the problem, we first establish the coordinates of the points in the unit square \\(ABCD\\). \n\nLet the square \\(ABCD\\) have the following coordinates:\n- \\(A(0, 1)\\)\n- \\(B(1, 1)\\)\n- \\(C(1, 0)\\)\n- \\(D(0, 0)\\)\n\nNext, we define point \\(P\\) on segment \\(AB\\) such that \\(AP = \\frac{1"
                ],
                [
                  "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 find the function \\( f \\) satisfying the conditions \\( f(f(f(n))) = 8n - 7 \\) and \\( f(2n) = 2f(n) + 1 \\), let's start investigating each condition.\n\nFrom the first condition, we can substitute \\( n = 1 \\):\n\\[\nf(f(f(1))) = 8 \\cdot 1 - 7 = 1.\n\\]\nLet \\( a = f(1) \\)."
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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,
                  "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-"
                ],
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                  "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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                  "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
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                  "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
                ],
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                  "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
                ],
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                  "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
                ],
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                  "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
                ],
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                  "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
                ],
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                  "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
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                  "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
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                  "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
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                  "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?",
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                  "There exists a unique increasing geometric sequence of five 2-digit positive integers. What is their sum?",
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                  "For how many positive integers $m$ does the equation $\\vert \\vert x-1 \\vert -2 \\vert=\\frac{m}{100}$ have $4$ distinct solutions?",
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                  "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
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                  "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?",
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                  "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.",
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                  "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 $f(100)$.",
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