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[{"content":"Suppose that $e$, $f$, $g$, and $h$ are positive integers such that $e^7 = f^5$, $g^4 =(...TRUNCATED) | "Suppose that $e$, $f$, $g$, and $h$ are positive integers such that $e^7 = f^5$, $g^4 = h^3$, and $(...TRUNCATED) | "Suppose that $e$, $f$, $g$, and $h$ are positive integers such that $e^7 = f^5$, $g^4 = h^3$, and $(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | "Given:\n\\[\ne^7 = f^5, \\quad g^4 = h^3, \\quad g - e = 24,\n\\]\nwhere \\(e, f, g, h\\) are posit(...TRUNCATED) | 3,448 | 18,903 | 256 | ["Okay, let's see. The problem involves four positive integers: e, f, g, and h. We have two equation(...TRUNCATED) | ["","","","Given the problem where \\( e \\), \\( f \\), \\( g \\), and \\( h \\) are positive integ(...TRUNCATED) | ["<think>\nOkay, let's see. The problem involves four positive integers: e, f, g, and h. We have two(...TRUNCATED) | [
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[{"content":"Let $f(x)$ be a function defined by $f(x) = x^2 + ax + b$, where $a$ and $b$ are consta(...TRUNCATED) | "Let $f(x)$ be a function defined by $f(x) = x^2 + ax + b$, where $a$ and $b$ are constants. If $f(2(...TRUNCATED) | "Let $f(x)$ be a function defined by $f(x) = x^2 + ax + b$, where $a$ and $b$ are constants. If $f(2(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | "Given the function:\n\\[\nf(x) = x^2 + ax + b,\n\\]\nwhere \\(a\\) and \\(b\\) are constants.\n\nWe(...TRUNCATED) | 5,819 | 11,197 | 2,039 | ["Okay, so I need to find the value of 'a' in the quadratic function f(x) = x² + a x + b. The given(...TRUNCATED) | ["Given the function \\( f(x) = x^2 + ax + b \\) with constants \\( a \\) and \\( b \\), we need to (...TRUNCATED) | ["<think>\nOkay, so I need to find the value of 'a' in the quadratic function f(x) = x² + a x + b. (...TRUNCATED) | [
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[{"content":"Find the total surface area of a tetrahedron whose faces are all rectangles with intege(...TRUNCATED) | "Find the total surface area of a tetrahedron whose faces are all rectangles with integer dimensions(...TRUNCATED) | "Find the total surface area of a tetrahedron whose faces are all rectangles with integer dimensions(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | "Let's analyze the problem carefully.\n\n---\n\n**Problem Restatement:**\n\nWe have a tetrahedron wi(...TRUNCATED) | 35,708 | 22,878 | 2,463 | ["Okay, so I need to find the total surface area of a tetrahedron where all the faces are rectangles(...TRUNCATED) | ["The problem requires finding the total surface area of a tetrahedron whose faces are all rectangle(...TRUNCATED) | ["<think>\nOkay, so I need to find the total surface area of a tetrahedron where all the faces are r(...TRUNCATED) | [
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] | "The provided answer consists of a lengthy, continuous block of text without any blank lines separat(...TRUNCATED) | ["<think>\nOkay, so I need to find the total surface area of a tetrahedron where all the faces are r(...TRUNCATED) | "<think>\nAlright, so I need to find the total surface area of a tetrahedron where all the faces are(...TRUNCATED) | [{"from":"human","value":"Find the total surface area of a tetrahedron whose faces are all rectangle(...TRUNCATED) |
[{"content":"With all angles measured in degrees, the product $\\prod_{k=1}^{30} \\sec^2(3k-2)^\\cir(...TRUNCATED) | "With all angles measured in degrees, the product $\\prod_{k=1}^{30} \\sec^2(3k-2)^\\circ=a^b$, wher(...TRUNCATED) | "With all angles measured in degrees, the product $\\prod_{k=1}^{30} \\sec^2(3k-2)^\\circ=a^b$, wher(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | "Given the product:\n\\[\n\\prod_{k=1}^{30} \\sec^2 (3k - 2)^\\circ\n\\]\nwhere all angles are measu(...TRUNCATED) | 27,565 | 20,740 | 2,819 | ["Okay, so I need to find the product of sec²(3k-2) degrees from k=1 to 30, and express it as a^b w(...TRUNCATED) | ["To solve the problem, we need to find the product \\(\\prod_{k=1}^{30} \\sec^2(3k-2)^\\circ\\) and(...TRUNCATED) | ["<think>\nOkay, so I need to find the product of sec²(3k-2) degrees from k=1 to 30, and express it(...TRUNCATED) | [
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] | "The provided answer, while lengthy in explanation, consists of numerous long paragraphs that exceed(...TRUNCATED) | ["<think>\nAlright, so I need to find the product of sec squared of angles from k=1 to 30, where eac(...TRUNCATED) | "<think>\nAlright, let's tackle this problem. So, we need to find the product of sec² terms from k=(...TRUNCATED) | [{"from":"human","value":"With all angles measured in degrees, the product $\\prod_{k=1}^{30} \\sec^(...TRUNCATED) |
[{"content":"For what values of the constant $k$ does the graph of $f(x) = \\frac{x^2-kx+3}{x^2+x-21(...TRUNCATED) | "For what values of the constant $k$ does the graph of $f(x) = \\frac{x^2-kx+3}{x^2+x-21}$ have exac(...TRUNCATED) | "For what values of the constant $k$ does the graph of $f(x) = \\frac{x^2-kx+3}{x^2+x-21}$ have exac(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | "We have the function\n\n\\[\nf(x) = \\frac{x^2 - kx + 3}{x^2 + x - 21}.\n\\]\n\nWe want to find for(...TRUNCATED) | 1,237 | 14,406 | 142 | ["Okay, so I need to figure out for which values of k the function f(x) = (x² - kx + 3)/(x² + x - (...TRUNCATED) | ["To determine the values of the constant \\( k \\) for which the graph of \\( f(x) = \\frac{x^2 - k(...TRUNCATED) | ["<think>\nOkay, so I need to figure out for which values of k the function f(x) = (x² - kx + 3)/(x(...TRUNCATED) | [
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] | "The provided answer contains no long paragraphs. Each section is either concise or broken into smal(...TRUNCATED) | ["<think>\nOkay, so I need to figure out for which values of k the function f(x) = (x² - kx + 3)/(x(...TRUNCATED) | "<think>\nOkay, so I need to figure out for which values of k the function f(x) = (x² - kx + 3)/(x(...TRUNCATED) | [{"from":"human","value":"For what values of the constant $k$ does the graph of $f(x) = \\frac{x^2-k(...TRUNCATED) |
[{"content":"Determine the value of the infinite product $$\\left(1+\\frac{2}{1}\\right)\\left(1+\\f(...TRUNCATED) | "Determine the value of the infinite product $$\\left(1+\\frac{2}{1}\\right)\\left(1+\\frac{2}{3}\\r(...TRUNCATED) | "Determine the value of the infinite product $$\\left(1+\\frac{2}{1}\\right)\\left(1+\\frac{2}{3}\\r(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | "We are asked to evaluate the infinite product\n\\[\n\\prod_{n=0}^\\infty \\left(1 + \\frac{2}{a_n}\(...TRUNCATED) | 3,479 | 12,948 | 250 | ["Okay, so I need to figure out the value of this infinite product: \n\n\\[\n\\left(1 + \\frac{2}{1}(...TRUNCATED) | ["","","","","The given infinite product is:\n\n$$\\left(1+\\frac{2}{1}\\right)\\left(1+\\frac{2}{3}(...TRUNCATED) | ["<think>\nOkay, so I need to figure out the value of this infinite product: \n\n\\[\n\\left(1 + \\f(...TRUNCATED) | [
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] | "The response does not contain any long paragraphs, as the text is broken down into smaller sections(...TRUNCATED) | ["<think>\nOkay, so I need to figure out the value of this infinite product: \n\n\\[\n\\left(1 + \\f(...TRUNCATED) | "<think>\nOkay, let's try to figure out the value of this infinite product. The product is given as:(...TRUNCATED) | [{"from":"human","value":"Determine the value of the infinite product $$\\left(1+\\frac{2}{1}\\right(...TRUNCATED) |
[{"content":"A quadrilateral $ABCD$ has $AB = 10$ and $CD = 6$. The diagonals meet at point $E$, an(...TRUNCATED) | "A quadrilateral $ABCD$ has $AB = 10$ and $CD = 6$. The diagonals meet at point $E$, and it is give(...TRUNCATED) | "A quadrilateral $ABCD$ has $AB = 10$ and $CD = 6$. The diagonals meet at point $E$, and it is give(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | "**Problem recap:**\n\nGiven a quadrilateral \\(ABCD\\) with:\n- \\(AB = 10\\)\n- \\(CD = 6\\)\n- Di(...TRUNCATED) | 38,564 | 14,863 | 3,146 | ["Okay, so I need to find the possible values of BC in this quadrilateral ABCD. Let me try to visual(...TRUNCATED) | ["Given a quadrilateral \\(ABCD\\) with \\(AB = 10\\), \\(CD = 6\\), and diagonals intersecting at p(...TRUNCATED) | ["<think>\nOkay, so I need to find the possible values of BC in this quadrilateral ABCD. Let me try (...TRUNCATED) | [
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[{"content":"In triangle $ABC$, the median $AM$ is perpendicular to the bisector $BD$ of angle $B$. (...TRUNCATED) | "In triangle $ABC$, the median $AM$ is perpendicular to the bisector $BD$ of angle $B$. If $AB = 15$(...TRUNCATED) | "In triangle $ABC$, the median $AM$ is perpendicular to the bisector $BD$ of angle $B$. If $AB = 15$(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | "**Given:**\n- Triangle \\( ABC \\) with sides \\( AB = 15 \\), \\( AC = 20 \\), \\( BC = 25 \\).\n-(...TRUNCATED) | 16,896 | 13,378 | 434 | ["Okay, so I need to find the length of segment MD in triangle ABC where the median AM is perpendicu(...TRUNCATED) | ["Given triangle \\(ABC\\) with \\(AB = 15\\), \\(AC = 20\\), and \\(BC = 25\\), we need to find the(...TRUNCATED) | ["<think>\nOkay, so I need to find the length of segment MD in triangle ABC where the median AM is p(...TRUNCATED) | [
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[{"content":"Find the sum of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$, given that one of the rea(...TRUNCATED) | "Find the sum of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$, given that one of the real roots is a(...TRUNCATED) | "Find the sum of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$, given that one of the real roots is a(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | "Given the equation:\n\\[\nx^4 - 4x^3 + 6x^2 - 4x = 2005,\n\\]\nwe are told that one of the real roo(...TRUNCATED) | 31,019 | 12,319 | 475 | ["Okay, so I need to find the sum of the nonreal roots of the equation \\(x^4 - 4x^3 + 6x^2 - 4x = 2(...TRUNCATED) | ["Given the equation \\(x^4 - 4x^3 + 6x^2 - 4x = 2005\\), we rewrite it in standard polynomial form (...TRUNCATED) | ["<think>\nOkay, so I need to find the sum of the nonreal roots of the equation \\(x^4 - 4x^3 + 6x^2(...TRUNCATED) | [
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[{"content":"Find the sum of all positive integers $n$ for which there exists a polynomial $P(x)$ wi(...TRUNCATED) | "Find the sum of all positive integers $n$ for which there exists a polynomial $P(x)$ with integer c(...TRUNCATED) | "Find the sum of all positive integers $n$ for which there exists a polynomial $P(x)$ with integer c(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | "**Problem:**\nFind the sum of all positive integers \\( n \\) for which there exists a polynomial \(...TRUNCATED) | 26,364 | 14,095 | 2,444 | ["Okay, so I need to find all positive integers n such that there exists a polynomial P(x) with inte(...TRUNCATED) | ["To solve the problem, we need to find all positive integers \\( n \\) for which there exists a pol(...TRUNCATED) | ["<think>\nOkay, so I need to find all positive integers n such that there exists a polynomial P(x) (...TRUNCATED) | [
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