statement stringlengths 51 1.23k | solution stringlengths 56 6.77k | answer int64 0 2.98B | grade float64 0 1 | model_solution_0 stringlengths 3.32k 173k | model_solution_1 stringlengths 2.79k 161k | model_solution_2 stringlengths 3.58k 148k | model_solution_3 stringlengths 3.52k 151k | model_solution_4 stringlengths 2.76k 173k | model_solution_5 stringlengths 2.76k 165k | model_solution_6 stringlengths 2.59k 150k | model_solution_7 stringlengths 3.24k 141k |
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"Find the number of quadruples \\((a, b, c, d)\\) of integers with absolute value at most \\(5\\) su(...TRUNCATED) | "Let \\(x = a + c\\), \\(y = a - c\\), \\(w = b + d\\), and \\(z = b - d\\). Then\n\n\\[\n\\left(w^{(...TRUNCATED) | 49 | 0.875 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) |
What is the earliest row in which the number \(2004\) may appear? | "The sequence of rows is as follows:\n\n\\[\n\\begin{array}{ccccccc}\n\\cdots & 0 & 0 & 1 & 0 & 0 & (...TRUNCATED) | 12 | 0.125 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) |
"Positive integers \\(a_{1}, a_{2}, \\ldots, a_{7}, b_{1}, b_{2}, \\ldots, b_{7}\\) satisfy \\(2 \\l(...TRUNCATED) | "Let \\(B = b_{1} b_{2} \\cdots b_{7} - 128\\). Since\n\n\\[\na_{1}^{b_{1} b_{2} \\cdots b_{7}} \\eq(...TRUNCATED) | 675 | 0.25 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) |
"Let \\( P(n) = (n-1^3)(n-2^3) \\ldots (n-40^3) \\) for positive integers \\( n \\). Suppose that \\(...TRUNCATED) | "We first investigate what primes divide \\( d \\). Notice that a prime \\( p \\) divides \\( P(n) \(...TRUNCATED) | 48 | 0.75 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) |
"How many permutations \\( p(n) \\) of \\(\\{1,2,3, \\ldots, 35\\}\\) satisfy \\( a \\mid b \\) impl(...TRUNCATED) | "We look at small numbers first. It is not hard to reason that \\(1, 2, 3, 4\\), and \\(5\\) must be(...TRUNCATED) | 48 | 0.5 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) |
"Find the number of points \\( P \\in \\mathbb{Z}^{2} \\) that satisfy the following two conditions:(...TRUNCATED) | "Notice that \\( 38^{2} + 24^{2} = 2020 \\). Let \\( P \\) have coordinates \\((38, y)\\), and label(...TRUNCATED) | 16 | 0.5 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) |
"Ten positive integers are arranged around a circle. Each number is one more than the greatest commo(...TRUNCATED) | "First, note that all the integers must be at least 2, because the greatest common divisor of any tw(...TRUNCATED) | 28 | 0.625 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) |
"The integers \\(1, 2, \\ldots, 64\\) are written in the squares of an \\(8 \\times 8\\) chessboard,(...TRUNCATED) | "We aim to maximize the sum of the numbers on the main diagonal (top left to bottom right). If we co(...TRUNCATED) | 432 | 0 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) |
"For how many ordered triplets \\((a, b, c)\\) of positive integers less than \\(10\\) is the produc(...TRUNCATED) | "Solution: One of the numbers must be \\(5\\). The other two numbers must have a product divisible b(...TRUNCATED) | 78 | 0 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) |
"Let \\(1, \\alpha_{1}, \\alpha_{2}, \\ldots, \\alpha_{10}\\) be the roots of the polynomial \\(x^{1(...TRUNCATED) | "The existence and uniqueness of this polynomial (up to sign) are assumed. Since all of the roots of(...TRUNCATED) | 15 | 0 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) |
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