problem string | solution string | answer string | subject string | level int64 | unique_id string | idx int64 | ete_solution string | gen_length int64 | num_steps int64 | pareto_index int64 | sampling_params string | worst_case_steps int64 | best_case_steps int64 | explore_failures int64 | explore_count int64 | solution_reward float64 | extracted_pred_answer string | extracted_gold_answer string | is_solution_correct bool | sequences list |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | For the piecewise function to be continuous, the cases must "meet" at $2$ and $-2$. For example, $ax+3$ and $x-5$ must be equal when $x=2$. This implies $a(2)+3=2-5$, which we solve to get $2a=-6 \Rightarrow a=-3$. Similarly, $x-5$ and $2x-b$ must be equal when $x=-2$. Substituting, we get $-2-5=2(-2)-b$, which implies... | 0 | Algebra | 5 | train/algebra/0.json | 0 | To ensure that the piecewise function \( f(x) \) is continuous, we need to ensure that the left-hand limit, right-hand limit, and the function value at the transition points are equal at \( x = -2 \) and \( x = 2 \).
First, let's find the left-hand limit at \( x = -2 \):
\[
\lim_{x \to -2^-} f(x) = 2(-2) - b = -4 - b
... | 512 | 55 | 8 | {"accumulative_error_tolerance": 0.2, "alpha": 0.86, "block_length": 64, "compute_log_probs": false, "confidence_decode": false, "continuous_explore_failure_tolerance": 3, "decode_mode": "anchor_latest", "dynamic_threshold": false, "explore_before_steps": 512.0, "explore_machanism": "current_block_only", "explore_mecha... | 75 | 45 | 0 | 10 | 1 | ['0', '0'] | ['0', '0'] | true | [
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... |
"Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le(...TRUNCATED) | "For the piecewise function to be continuous, the cases must \"meet\" at $2$ and $-2$. For example, (...TRUNCATED) | 0 | Algebra | 5 | train/algebra/0.json | 0 | "To ensure that the piecewise function \\( f(x) \\) is continuous, we need to ensure that the left-h(...TRUNCATED) | 512 | 58 | 9 | "{\"accumulative_error_tolerance\": 0.2, \"alpha\": 0.86, \"block_length\": 64, \"compute_log_probs\(...TRUNCATED) | 78 | 48 | 0 | 10 | 1 | ['0', '0'] | ['0', '0'] | true | [[126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,(...TRUNCATED) |
"Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le(...TRUNCATED) | "For the piecewise function to be continuous, the cases must \"meet\" at $2$ and $-2$. For example, (...TRUNCATED) | 0 | Algebra | 5 | train/algebra/0.json | 0 | "To ensure that the piecewise function \\( f(x) \\) is continuous, we need to ensure that the left-h(...TRUNCATED) | 512 | 61 | 11 | "{\"accumulative_error_tolerance\": 0.2, \"alpha\": 0.86, \"block_length\": 64, \"compute_log_probs\(...TRUNCATED) | 80 | 50 | 0 | 10 | 1 | ['0', '0'] | ['0', '0'] | true | [[126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,(...TRUNCATED) |
"Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le(...TRUNCATED) | "For the piecewise function to be continuous, the cases must \"meet\" at $2$ and $-2$. For example, (...TRUNCATED) | 0 | Algebra | 5 | train/algebra/0.json | 0 | "To ensure that the piecewise function \\( f(x) \\) is continuous, we need to ensure that the left-h(...TRUNCATED) | 512 | 55 | 13 | "{\"accumulative_error_tolerance\": 0.2, \"alpha\": 0.86, \"block_length\": 64, \"compute_log_probs\(...TRUNCATED) | 75 | 45 | 0 | 10 | 1 | ['0', '0'] | ['0', '0'] | true | [[126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,(...TRUNCATED) |
"Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le(...TRUNCATED) | "For the piecewise function to be continuous, the cases must \"meet\" at $2$ and $-2$. For example, (...TRUNCATED) | 0 | Algebra | 5 | train/algebra/0.json | 0 | "To ensure that the piecewise function \\( f(x) \\) is continuous, we need to ensure that the left-h(...TRUNCATED) | 512 | 58 | 14 | "{\"accumulative_error_tolerance\": 0.2, \"alpha\": 0.86, \"block_length\": 64, \"compute_log_probs\(...TRUNCATED) | 78 | 48 | 0 | 10 | 1 | ['0', '0'] | ['0', '0'] | true | [[126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,(...TRUNCATED) |
"Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le(...TRUNCATED) | "For the piecewise function to be continuous, the cases must \"meet\" at $2$ and $-2$. For example, (...TRUNCATED) | 0 | Algebra | 5 | train/algebra/0.json | 0 | "To ensure that the piecewise function \\( f(x) \\) is continuous, we need to make sure that the val(...TRUNCATED) | 512 | 97 | 23 | "{\"accumulative_error_tolerance\": 0.2, \"alpha\": 0.86, \"block_length\": 64, \"compute_log_probs\(...TRUNCATED) | 117 | 87 | 0 | 10 | 1 | ['0', '0'] | ['0', '0'] | true | [[126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,(...TRUNCATED) |
"Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le(...TRUNCATED) | "For the piecewise function to be continuous, the cases must \"meet\" at $2$ and $-2$. For example, (...TRUNCATED) | 0 | Algebra | 5 | train/algebra/0.json | 0 | "To ensure that the piecewise function \\( f(x) \\) is continuous, we need to make sure that the val(...TRUNCATED) | 512 | 97 | 24 | "{\"accumulative_error_tolerance\": 0.2, \"alpha\": 0.86, \"block_length\": 64, \"compute_log_probs\(...TRUNCATED) | 117 | 87 | 0 | 10 | 1 | ['0', '0'] | ['0', '0'] | true | [[126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,(...TRUNCATED) |
"Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le(...TRUNCATED) | "For the piecewise function to be continuous, the cases must \"meet\" at $2$ and $-2$. For example, (...TRUNCATED) | 0 | Algebra | 5 | train/algebra/0.json | 0 | "To ensure that the piecewise function \\( f(x) \\) is continuous, we need to make sure that the val(...TRUNCATED) | 512 | 101 | 25 | "{\"accumulative_error_tolerance\": 0.2, \"alpha\": 0.86, \"block_length\": 64, \"compute_log_probs\(...TRUNCATED) | 121 | 91 | 0 | 10 | 1 | ['0', '0'] | ['0', '0'] | true | [[126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,(...TRUNCATED) |
"Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le(...TRUNCATED) | "For the piecewise function to be continuous, the cases must \"meet\" at $2$ and $-2$. For example, (...TRUNCATED) | 0 | Algebra | 5 | train/algebra/0.json | 0 | "To ensure that the piecewise function \\( f(x) \\) is continuous, we need to make sure that the val(...TRUNCATED) | 512 | 132 | 33 | "{\"accumulative_error_tolerance\": 0.2, \"alpha\": 0.86, \"block_length\": 64, \"compute_log_probs\(...TRUNCATED) | 152 | 122 | 0 | 10 | 1 | ['0', '0'] | ['0', '0'] | true | [[126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,(...TRUNCATED) |
Twenty-seven increased by twice a number is 39. What is the number? | "We are asked to solve $27+2x=39$ for $x$. We subtract 27 from both sides and then multiply both si(...TRUNCATED) | 6 | Prealgebra | 1 | train/prealgebra/1915.json | 2 | "To solve the problem, let's denote the unknown number by \\( x \\).\n\nStep 1: TranslateTwenty-seve(...TRUNCATED) | 512 | 20 | 0 | "{\"accumulative_error_tolerance\": 0.2, \"alpha\": 0.86, \"block_length\": 64, \"compute_log_probs\(...TRUNCATED) | 28 | 16 | 0 | 4 | 1 | ['6', '6'] | ['6', '6'] | true | [[126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,126336,(...TRUNCATED) |
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