statement stringlengths 118 1.2k | solution stringlengths 229 4.09k | answer int64 1 310M | grade float64 0 1 | short_summary stringlengths 50 268 | model_solution_0 stringlengths 5.97k 78.2k | model_solution_1 stringlengths 5.85k 76.9k | model_solution_2 stringlengths 5.65k 77.8k | model_solution_3 stringlengths 4.66k 76.8k | model_solution_4 stringlengths 5.42k 72.2k | model_solution_5 stringlengths 4.88k 78.1k | model_solution_6 stringlengths 4.99k 78.3k | model_solution_7 stringlengths 4.06k 77.3k |
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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 | 0.5 | start/end differ;two cases;remove endpoints;binomial count;double by symmetry | "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 | Factor with r;Power of D;Isosceles identification;Angle-chase for α;Central–inscribed relation | "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 | "Assume minimal;Inspect smallest cubes;Place 5th WLOG;Forces double elimination;Analyze sixth unique(...TRUNCATED) | "Okay, let's try to tackle this problem step by step. The problem is asking for the smallest positiv(...TRUNCATED) | "Okay, let's try to figure out this problem step by step. The question is asking for the smallest po(...TRUNCATED) | "Okay, let's try to figure out this problem. The question is asking for the smallest positive intege(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "Okay, let's try to figure out this problem step by step. The question is asking for the smallest po(...TRUNCATED) | "Okay, let's try to figure out this problem step by step. The question is asking for the smallest po(...TRUNCATED) | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | "Okay, let's try to figure out this problem step by step. The question is asking for the smallest po(...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 | 0.25 | Harmonic quadruple;Polar through O;Harmonic on BC;Solve harmonic equation | "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 | 0.125 | "GB3 is diameter;Swap A2/A3 B1/B5;Chords meet GB3;Angle equals ∠GG2B1;Half difference arcs;Substit(...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) | "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 | 0.75 | "Students as subsets;Define f = S Δ T;Fixed-point-free involution;Partition into pairs;Pairs: canno(...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) | "Okay, let's try to tackle this problem step by step using the given hints. First, let's restate the(...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 | 0.5 | "Pigeonhole: adjacent positives;Alternating sign inequalities;Multiply around cycle;Explicit constru(...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) | "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 | "convert to cot;cot double-angle;reciprocal → tan;split as 45°+θ;tan double-angle;common denomin(...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) | "Okay, let's try to solve this problem step by step. The problem is to compute \\(\\arctan \\left(\\(...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) | "Okay, let's try to solve this problem step by step. The problem is to compute \\(\\arctan \\left(\\(...TRUNCATED) | "Okay, let's try to solve this problem step by step. The problem is to compute arctan(tan 65° - 2 t(...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 | ED = AD;AED half-supplement;AED = BEC;AEB via full-angle | "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) | "Okay, let's try to solve this problem step by step. First, let's set up a coordinate system to mode(...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) | "Okay, let's try to solve this problem step by step. First, let's set up a coordinate system to mode(...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 | 1 | "Parity invariant;Even difference determines target;Each exchange reduces total;Balance then convert(...TRUNCATED) | "Okay, let's try to work through this problem step by step. The problem is about Bassanio having thr(...TRUNCATED) | "Okay, let's try to figure out this problem step by step. So, Bassanio has three red (R), four yello(...TRUNCATED) | "Okay, let's try to figure out this problem step by step. So, Bassanio has three red (R), four yello(...TRUNCATED) | "Okay, let's try to figure out this problem step by step. So, Bassanio has 3 red, 4 yellow, and 5 bl(...TRUNCATED) | "Okay, let's try to figure out this problem step by step. So, Bassanio has three red (R), four yello(...TRUNCATED) | "Okay, let's try to work through this problem step by step. So, the problem is about Bassanio having(...TRUNCATED) | "Okay, let's try to work through this problem step by step. First, let me make sure I understand the(...TRUNCATED) | "Okay, let's try to figure out this problem step by step. So, Bassanio has three red (R), four yello(...TRUNCATED) |
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