statement stringlengths 118 1.2k | solution stringlengths 229 4.09k | answer int64 1 310M | grade float64 0 1 | model_solution_0 stringlengths 6k 159k | model_solution_1 stringlengths 5.03k 126k | model_solution_2 stringlengths 5.18k 119k | model_solution_3 stringlengths 12.4k 115k | model_solution_4 stringlengths 5.23k 111k | model_solution_5 stringlengths 3.49k 124k | model_solution_6 stringlengths 12.5k 134k | model_solution_7 stringlengths 6.04k 127k |
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"How many ways are there for Nick to travel from \\((0,0)\\) to \\((16,16)\\) in the coordinate plan(...TRUNCATED) | "This condition is equivalent to the first and last step being in different directions, as if you sw(...TRUNCATED) | 310,235,040 | 1 | "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 \\( \\triangle ABC \\) be an obtuse triangle with circumcenter \\( O \\) such that \\( \\angle (...TRUNCATED) | "Let the radius of the circumcircle of \\( \\triangle ABC \\) be \\( r \\).\n\n\\[\n\\begin{align*}\(...TRUNCATED) | 35 | 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) |
"Problem: What is the smallest positive integer representable as the sum of the cubes of three posit(...TRUNCATED) | "Solution: Let this smallest positive integer be represented as \\(a^{3}+b^{3}+c^{3}=d^{3}+e^{3}+f^{(...TRUNCATED) | 251 | 1 | "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 \\( \\triangle ABC \\) be a triangle with \\( BC = 2007 \\), \\( CA = 2008 \\), \\( AB = 2009 \(...TRUNCATED) | "Let line \\( AD \\) meet \\( \\omega \\) again at \\( H \\). Since \\( AF \\) and \\( AE \\) are ta(...TRUNCATED) | 2,014,024 | 1 | "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 \\( G, A_{1}, A_{2}, A_{3}, A_{4}, B_{1}, B_{2}, B_{3}, B_{4}, B_{5} \\) be ten points on a cir(...TRUNCATED) | "Note that \\( G B_{3} \\) is a diameter of the circle. As a result, \\( A_{2}, A_{3} \\) are symmet(...TRUNCATED) | 12 | 1 | "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) |
"At a certain college, there are $10$ clubs and some number of students. For any two different stude(...TRUNCATED) | "Let \\( C \\) be the set of clubs; each student then corresponds to a subset of \\( C \\) (the club(...TRUNCATED) | 513 | 1 | "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 \\( x_{1}, x_{2}, \\ldots, x_{2022} \\) be nonzero real numbers. Suppose that \\( x_{k}+\\frac{(...TRUNCATED) | "Let the answer be \\( M \\). If \\( M > 1011 \\), there would exist two consecutive positive terms (...TRUNCATED) | 1,010 | 1 | "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) |
"Compute \\(\\arctan \\left(\\tan 65^{\\circ}-2 \\tan 40^{\\circ}\\right)\\). (Express your answer i(...TRUNCATED) | "First Solution: We have\n\\[\n\\tan 65^{\\circ}-2 \\tan 40^{\\circ}=\\cot 25^{\\circ}-2 \\cot 50^{\(...TRUNCATED) | 25 | 1 | "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) |
"Rectangle \\(ABCD\\) has sides in the ratio of \\(\\sqrt{2}\\) to \\(1\\). If \\(\\triangle DEC\\) (...TRUNCATED) | "Since \\(\\triangle DEC\\) is an isosceles right triangle, we have \\(ED = CD = \\sqrt{2} \\cdot AD(...TRUNCATED) | 135 | 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) |
"Bassanio has three red coins, four yellow coins, and five blue coins. At any point, he may give Shy(...TRUNCATED) | "Let \\( r, y, b \\) denote the numbers of red, yellow, and blue coins respectively. Note that each (...TRUNCATED) | 7 | 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) |
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