sample_index int64 | problem_id int64 | problem string | ground_truth string | partial_solution string | step_number int64 | total_steps int64 | pct_complete float64 | empirical_success_rate float64 | num_continuations int64 | num_correct int64 | full_initial_generation string | continuations list | parsed_answers list | initial_intervention_used bool | continuation_interventions list | num_continuations_intervened int64 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients $2$ and $-2,$ respectively. The graphs of both polynomials pass through the two points $(16,54)$ and $(20,53).$ Find $P(0) + Q(0).$ | 116 | This is a complex or challenging question, and it is difficult to provide a direct and correct answer. I need to think about it.
Well, so we have two quadratic polynomials, P(x) and Q(x). Both have leading coefficients 2 and -2, respectively. That means I can write them out with their general forms first. Let's start b... | 1 | 43 | 0.023256 | 0.9 | 10 | 9 | This is a complex or challenging question, and it is difficult to provide a direct and correct answer. I need to think about it.
Well, so we have two quadratic polynomials, P(x) and Q(x). Both have leading coefficients 2 and -2, respectively. That means I can write them out with their general forms first. Let's start b... | [
"This is a problem about quadratic polynomials with given leading coefficients and shared points. Let's try to formalize this.\n\nFirst, since P(x) is quadratic with leading coefficient 2, it should be P(x) = 2x² + bx + c, where b and c are constants we need to find. Similarly, Q(x) is quadratic with leading coeffi... | [
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0 | 0 | Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients $2$ and $-2,$ respectively. The graphs of both polynomials pass through the two points $(16,54)$ and $(20,53).$ Find $P(0) + Q(0).$ | 116 | This is a complex or challenging question, and it is difficult to provide a direct and correct answer. I need to think about it.
Well, so we have two quadratic polynomials, P(x) and Q(x). Both have leading coefficients 2 and -2, respectively. That means I can write them out with their general forms first. Let's start b... | 21 | 43 | 0.488372 | 1 | 10 | 10 | This is a complex or challenging question, and it is difficult to provide a direct and correct answer. I need to think about it.
Well, so we have two quadratic polynomials, P(x) and Q(x). Both have leading coefficients 2 and -2, respectively. That means I can write them out with their general forms first. Let's start b... | [
"Okay, so the problem is about two quadratic polynomials, P(x) and Q(x). P has leading coefficient 2, Q has leading coefficient -2. Both pass through (16, 54) and (20, 53). Need to find P(0) + Q(0).\n\nFirst, I should write down what P(x) and Q(x) look like. Since they're quadratics with those leading coefficients,... | [
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0 | 0 | "Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients $2$ and $-2,$ respectively. The g(...TRUNCATED) | 116 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | 42 | 43 | 0.976744 | 1 | 10 | 10 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | ["Okay, so I need to solve this problem where there are two quadratic polynomials P(x) and Q(x) with(...TRUNCATED) | [
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1 | 1 | "Three spheres with radii $11$, $13$, and $19$ are mutually externally tangent. A plane intersects t(...TRUNCATED) | 756 | "Okay, let's try to solve this problem step by step. First, I need to recall some geometry about sph(...TRUNCATED) | 1 | 34 | 0.029412 | 1 | 10 | 10 | "Okay, let's try to solve this problem step by step. First, I need to recall some geometry about sph(...TRUNCATED) | ["This is a complex or challenging question, and it is difficult to provide a direct and correct ans(...TRUNCATED) | ["756","756","756","756","756","756\n\nBased on this reasoning, the answer is <answer>756","756","75(...TRUNCATED) | false | [
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1 | 1 | "Three spheres with radii $11$, $13$, and $19$ are mutually externally tangent. A plane intersects t(...TRUNCATED) | 756 | "Okay, let's try to solve this problem step by step. First, I need to recall some geometry about sph(...TRUNCATED) | 17 | 34 | 0.5 | 1 | 10 | 10 | "Okay, let's try to solve this problem step by step. First, I need to recall some geometry about sph(...TRUNCATED) | ["Okay, let's check that again. We found d₁ = 4, d₂ = 8, d₃ = 16. Let's verify k²: 121 - 16 =(...TRUNCATED) | [
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1 | 1 | "Three spheres with radii $11$, $13$, and $19$ are mutually externally tangent. A plane intersects t(...TRUNCATED) | 756 | "Okay, let's try to solve this problem step by step. First, I need to recall some geometry about sph(...TRUNCATED) | 33 | 34 | 0.970588 | 1 | 10 | 10 | "Okay, let's try to solve this problem step by step. First, I need to recall some geometry about sph(...TRUNCATED) | ["<answer>756</answer>","Okay, let's see. I need to solve this problem about three spheres and a pla(...TRUNCATED) | [
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2 | 2 | "Let $ABCD$ be a parallelogram with $\\angle BAD < 90^\\circ.$ A circle tangent to sides $\\overline(...TRUNCATED) | 150 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | 1 | 105 | 0.009524 | 0 | 10 | 0 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | ["This is a complex problem, but let's try to break it down step by step. First, maybe I should set (...TRUNCATED) | [
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2 | 2 | "Let $ABCD$ be a parallelogram with $\\angle BAD < 90^\\circ.$ A circle tangent to sides $\\overline(...TRUNCATED) | 150 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | 53 | 105 | 0.504762 | 0 | 10 | 0 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | ["Okay, let's start solving this problem step by step as instructed. First, I need to set up a coord(...TRUNCATED) | [null,null,null,null,null,null,null,null,"... but wait, no, I need to do the steps. Let's continue.\(...TRUNCATED) | true | [
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2 | 2 | "Let $ABCD$ be a parallelogram with $\\angle BAD < 90^\\circ.$ A circle tangent to sides $\\overline(...TRUNCATED) | 150 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | 104 | 105 | 0.990476 | 0.1 | 10 | 1 | "This is a complex or challenging question, and it is difficult to provide a direct and correct answ(...TRUNCATED) | ["Okay, let's recap with the simplified center. Center is (6, q), radius q. The circle is tangent to(...TRUNCATED) | [null,null,null,null,null,null,"... but wait, no, I need to do the steps. Let's continue.\n\nSo expa(...TRUNCATED) | true | [
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3 | 3 | "For any finite set $X$, let $| X |$ denote the number of elements in $X$. Define\n\\[S_n = \\sum | (...TRUNCATED) | 245 | "Okay, let's try to solve this problem step by step. First, I need to understand what S_n is. The pr(...TRUNCATED) | 1 | 78 | 0.012821 | 0.5 | 10 | 5 | "Okay, let's try to solve this problem step by step. First, I need to understand what S_n is. The pr(...TRUNCATED) | ["This is a bit complex, but maybe I can use linearity of expectation or linearity of summation by c(...TRUNCATED) | ["(2*2022*4041)/2021². Wait, let's compute this fraction.\n\nLet's denote m = 2021, so n = 2022 = m(...TRUNCATED) | true | [
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