question stringlengths 51 786 | answer stringlengths 73 4.13k | input_ids listlengths 1.61k 30.1k | reply stringlengths 4.13k 91.7k | original_index int64 732 913 | claims listlengths 36 1.36k |
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"Let $r$ be a complex number such that $r^5 = 1$ and $r \\neq 1.$ Compute\n\\[(r - 1)(r^2 - 1)(r^3 (...TRUNCATED) | "We can write $r^5 - 1 = 0,$ which factors as\n\\[(r - 1)(r^4 + r^3 + r^2 + r + 1) = 0.\\]Since $r \(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to compute the product (r - 1)(r² - 1)(r³ - 1)(r⁴ - 1) where r is a co(...TRUNCATED) | 732 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29(...TRUNCATED) |
"Find the number of ordered pairs of integers $(a,b)$ with $1 \\leq a \\leq 100$ and $b \\geq 0$ suc(...TRUNCATED) | "Because the given quadratic has leading coefficient $1$, both factors must be of the form $x-c$ (or(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to find the number of ordered pairs (a, b) where 1 ≤ a ≤ 100 and b ≥(...TRUNCATED) | 733 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29(...TRUNCATED) |
"The four complex roots of\n\\[2z^4 + 8iz^3 + (-9 + 9i)z^2 + (-18 - 2i)z + (3 - 12i) = 0,\\]when plo(...TRUNCATED) | "Let $a,$ $b,$ $c,$ $d$ be the roots of the quartic. Let $A$ be the point corresponding to complex (...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to find the area of a rhombus formed by the four complex roots of this qua(...TRUNCATED) | 734 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],"cla(...TRUNCATED) |
"Find all real numbers $a$ such that the equation\n\\[x^3 - ax^2 - 2ax + a^2 - 1 = 0\\]has exactly o(...TRUNCATED) | "Writing the equation as a quadratic in $a,$ we get\n\\[a^2 - (x^2 + 2x) a + (x^3 - 1) = a^2 - (x^2 (...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to find all real numbers $ a $ such that the equation $ x^3 - a x^2 - 2a x(...TRUNCATED) | 735 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29(...TRUNCATED) |
"Let $z$ be a complex number such that $|z - 5 - i| = 5.$ Find the minimum value of\n\\[|z - 1 + 2i(...TRUNCATED) | "Let $z = x + yi,$ where $x$ and $y$ are real numbers. Then $|x + yi - 5 - i| = |(x - 5) + (y - 1)i(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to find the minimum value of the expression |z - 1 + 2i|² + |z - 9 - 4i|(...TRUNCATED) | 736 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29(...TRUNCATED) |
"Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number. Two of the roo(...TRUNCATED) | "By Factor Theorem,\n\\[f(x) = (x - r - 1)(x - r - 7)(x - a)\\]and\n\\[g(x) = (x - r - 3)(x - r - 9)(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, let's try to solve this problem. So, we have two monic cubic polynomials, f(x) and g(...TRUNCATED) | 737 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11],"claim_text":"\nOkay, let's try to solve this proble(...TRUNCATED) |
"The injective function $f(x)$ satisfies\n\\[f(x) f(x + y) = f(2x + y) - xf(x + y) + x\\]for all rea(...TRUNCATED) | "Setting $x = y = 0$ in the given functional equation, we get\n\\[f(0)^2 = f(0),\\]so $f(0) = 0$ or (...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to find an injective function f(x) that satisfies the equation:\n\nf(x) * (...TRUNCATED) | 738 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29(...TRUNCATED) |
"Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for(...TRUNCATED) | "Let $a$ and $b$ be the integer roots. Then we can write \\[f(x) = k(x-a)(x-b)\\]for some integer $k(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to find the number of second-degree polynomials f(x) with integer coeffici(...TRUNCATED) | 739 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29(...TRUNCATED) |
"Find the number of ordered pairs $(x,y)$ of real numbers that satisfy\n\\[x^2 + 2y^2 + \\frac{1}{2}(...TRUNCATED) | "Moving all the terms to the left-hand side, we get\n\\[x^2 - 2xy + 2y^2 - x + \\frac{1}{2} \\le 0.\(...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to find the number of ordered pairs (x, y) of real numbers that satisfy th(...TRUNCATED) | 740 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29(...TRUNCATED) |
"Let $x = \\sqrt{\\frac{\\sqrt{53}}{2} + \\frac{3}{2}}.$ There exist unique positive integers $a,$ (...TRUNCATED) | "We have that $x^2 = \\frac{\\sqrt{53}}{2} + \\frac{3}{2}.$ Then $2x^2 = \\sqrt{53} + 3,$ so $2x^2 (...TRUNCATED) | [151644,872,198,16141,279,2701,3405,13,10751,1469,697,4226,304,366,9217,1472,9217,29816,27,14582,268(...TRUNCATED) | "<think>\nOkay, so I need to find the value of a + b + c given this equation involving x raised to d(...TRUNCATED) | 741 | [{"aligned_token_ids":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],"claim_(...TRUNCATED) |
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