id int64 0 7.5k | problem stringlengths 16 4.31k | gt_answer stringlengths 0 421 | selected_response stringlengths 15 6.63k | selected_prm_score sequencelengths 1 39 | selected_heuristic_prm_score float64 0 1 | selected_prm_verification_score int64 0 1 | selected_gpt4o_verification_score stringclasses 3
values | selected_gpt4o_reasoning stringlengths 15 4.16k | responses sequencelengths 32 32 | prm_scores sequencelengths 32 32 | heuristic_prm_scores sequencelengths 32 32 | prm_verification_scores sequencelengths 32 32 | gpt4o_verification_scores sequencelengths 32 32 | gpt4o_reasonings sequencelengths 32 32 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 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). | 0 | ## Step 1: For the piecewise function to be continuous, the different parts must connect smoothly at the transition points. In this case, the transition points are $x=2$ and $x=-2$.
## Step 2: To ensure continuity at $x=2$, we need to make sure that the values of $ax+3$ and $x-5$ are equal at $x=2$. So, we set $a(2)... | [
0.98828125,
0.98828125,
0.99609375,
0.99609375,
1,
1,
1,
1
] | 0.996082 | 1 | 1 | ### Verification of Steps:
## Step 1:
Correct. The piecewise function must be continuous at the transition points $x=2$ and $x=-2$.
## Step 2:
Correct. To ensure continuity at $x=2$, we need $ax+3$ and $x-5$ to be equal at $x=2$. Setting $a(2) + 3 = 2 - 5$ is correct.
## Step 3:
Correct. Solving $2a + 3 = -3$ giv... | [
"## Step 1: To ensure continuity, we need to find the value of a and b such that the function is continuous at x=2 and x=-2.\nThe function is continuous if the left and right limits equal the function value at the point of discontinuity.\n\n## Step 2: First, let's evaluate the function at x=2 from the right side (u... | [
[
0.98046875,
0.953125,
0.98828125,
0.99609375,
1,
0.9921875,
0.99609375,
0.99609375,
1,
1,
1
],
[
0.95703125
],
[
0.96875,
0.97265625,
0.98828125,
0.9921875,
0.98828125,
0.9921875,
0.9921875,
0.9453125,
0.99609375,
1
... | [
0.9910307787273983,
0,
0.9834649982187614,
0.9960822307930701,
0.9941024520664549,
0,
0,
0.9848307991221487,
0.9926225041577527,
0,
0.9896841670016142,
0.9736001897768011,
0.9862582954992511,
0.974784519728217,
0,
0.9645889003597301,
0.9937731983680084,
0.9847461891224145,
0.9858... | [
1,
0,
1,
1,
1,
0,
0,
1,
1,
0,
1,
1,
1,
1,
0,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
0,
1,
1,
1,
1
] | [
"1",
"-1",
"1",
"1",
"-1",
"-1",
"-1",
"1",
"1",
"-1",
"0",
"1",
"1",
"1",
"-1",
"1",
"1",
"1",
"0",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"-1",
"-1",
"1",
"1",
"1",
"1"
] | [
"## Step-by-Step Verification:\n\n### Step 1:\nTo ensure continuity, we need to find the value of \\(a\\) and \\(b\\) such that the function is continuous at \\(x=2\\) and \\(x=-2\\).\n- This step is correct. For the function to be continuous, the left and right limits at the points of discontinuity must be equal.\... |
1 | "A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ an(...TRUNCATED) | 98 | "## Step 1: Define the initial formation\nLet's define the initial rectangular formation with $m$ me(...TRUNCATED) | [0.96875,0.98046875,0.9921875,0.9921875,0.98828125,0.9765625,0.98046875,0.94140625,0.96875,0.9921875(...TRUNCATED) | 0.981 | 1 | 1 | "## Verification of Steps:\n\n### Step 1: Define the initial formation\nThe initial formation is def(...TRUNCATED) | ["## Step 1: Define the given conditions\nLet's assume the original formation has $m$ members in eac(...TRUNCATED) | [[0.96875,0.98046875,0.96875,0.98828125,0.9765625,0.98046875,0.92578125,0.90625,0.984375,0.98828125,(...TRUNCATED) | [0.9687366071614697,0.9512762293135719,0.7324566069904447,0.9810004170307877,0.0,0.9110223813731235,(...TRUNCATED) | [
1,
1,
0,
1,
0,
1,
1,
1,
1,
1,
1,
1,
0,
0,
1,
0,
0,
0,
1,
0,
1,
0,
1,
1,
0,
0,
0,
1,
0,
0,
1,
0
] | ["1","1","0","1","-1","1","-1","1","0","1","1","1","-1","-1","1","0","-1","-1","1","0","1","-1","-1"(...TRUNCATED) | ["## Step-by-Step Verification:\n\n### Step 1: Define the given conditions\n- The original formation(...TRUNCATED) |
2 | What is the degree of the polynomial $(4 +5x^3 +100 +2\pi x^4 + \sqrt{10}x^4 +9)$? | 4 | "## Step 1: Identify the term with the highest exponent\nThe given polynomial is $4 +5x^3 +100 +2\\p(...TRUNCATED) | [
0.99609375,
0.99609375,
1,
1
] | 0.998045 | 1 | 1 | "## Step 1: Identify the term with the highest exponent\nThe given polynomial is $4 +5x^3 +100 +2\\p(...TRUNCATED) | ["## Step 1: Identify the highest power of x in the polynomial\nThe highest power of x in the given (...TRUNCATED) | [[0.95703125,0.97265625,0.9921875],[0.98828125,0.9921875,0.99609375],[0.9921875,0.99609375,0.9960937(...TRUNCATED) | [0.9738522770288571,0.9921823736611778,0.9951157480845482,0.9934878782902422,0.9960906862556632,0.99(...TRUNCATED) | [
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
0,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1
] | ["1","-1","1","1","-1","1","1","-1","-1","1","-1","1","-1","1","1","1","1","-1","1","-1","-1","-1","(...TRUNCATED) | ["## Step 1: Identify the highest power of x in the polynomial\nThe highest power of x in the given (...TRUNCATED) |
3 | Evaluate $\left\lceil3\left(6-\frac12\right)\right\rceil$. | 17 | "## Step 1: Calculate the expression inside the parentheses\nCalculate $6 - \\frac{1}{2} = 6 - 0.5 =(...TRUNCATED) | [
0.99609375,
1,
1,
1
] | 0.999022 | 1 | 1 | "Let's verify each step to ensure the correctness of the solution.\n\n## Step 1: Calculate the expre(...TRUNCATED) | ["## Step 1: Evaluate the expression inside the parentheses\n$6 - \\frac{1}{2} = 6 - 0.5 = 5.5$\n\n#(...TRUNCATED) | [[0.9921875,1.0,1.0,1.0],[0.9921875,1.0,1.0,1.0],[0.9921875,1.0,1.0,1.0],[0.9921875,1.0,1.0,1.0],[0.(...TRUNCATED) | [0.9980411267361815,0.9980411267361815,0.9980411267361815,0.9980411267361815,0.9980411267361815,0.99(...TRUNCATED) | [
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1
] | ["1","1","-1","1","-1","-1","1","1","-1","1","-1","-1","1","-1","1","1","1","-1","1","1","1","1","1"(...TRUNCATED) | ["## Step 1: Evaluate the expression inside the parentheses\n$6 - \\frac{1}{2} = 6 - 0.5 = 5.5$\n\nV(...TRUNCATED) |
4 | "Sam is hired for a 20-day period. On days that he works, he earns $\\$$60. For each day that he doe(...TRUNCATED) | 6 | "## Step 1: Let's denote the number of days Sam worked as W and the number of days he did not work a(...TRUNCATED) | [
1,
1,
1,
1,
1,
1,
1,
1
] | 1 | 1 | 1 | "## Step 1: Let's denote the number of days Sam worked as W and the number of days he did not work a(...TRUNCATED) | ["## Step 1: Define the variables and the equation representing the total earnings\nLet's define x a(...TRUNCATED) | [[0.9921875,1.0,1.0,1.0,1.0],[0.99609375,1.0,1.0,1.0,1.0,1.0,1.0],[0.99609375,0.95703125,0.99609375,(...TRUNCATED) | [0.9984325941733988,0.9994410278087485,0.9936712384561593,0.9992175264280057,1.0,0.9990220037201167,(...TRUNCATED) | [
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1
] | ["1","1","1","1","-1","-1","-1","-1","-1","1","1","-1","-1","1","1","1","-1","1","1","1","-1","1","1(...TRUNCATED) | ["## Step-by-Step Verification:\n\n### Step 1: Define the variables and the equation representing th(...TRUNCATED) |
5 | Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. | (3, -1) | "## Step 1: Rewrite the equation in standard form\nTo find the center of the circle, we need to rewr(...TRUNCATED) | [
1,
1,
1,
1,
1,
1,
1,
1
] | 1 | 1 | 1 | "## Verification of Steps:\n\n### Step 1: Rewrite the equation in standard form\nThe goal is to rewr(...TRUNCATED) | ["## Step 1: Convert the given equation to standard circle form\nThe equation can be rewritten as $x(...TRUNCATED) | [[0.9921875,1.0,0.99609375],[0.9921875,1.0,1.0,0.99609375],[0.99609375,1.0,1.0],[1.0,0.99609375,0.97(...TRUNCATED) | [0.9960886437646739,0.997065046227063,0.9986962175567516,0.9947564083312838,0.9992175264280057,0.995(...TRUNCATED) | [
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
0,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1
] | ["1","1","1","1","1","1","1","1","1","1","1","1","1","1","1","1","0","1","-1","1","-1","1","1","1","(...TRUNCATED) | ["## Step 1: Convert the given equation to standard circle form\nThe equation can be rewritten as $x(...TRUNCATED) |
6 | "What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q(...TRUNCATED) | [0,3) | not_gt_verified | [
0.0673828125
] | 0 | 0 | -1 | not_gt_verified | ["not_gt_verified","not_gt_verified","not_gt_verified","not_gt_verified","not_gt_verified","not_gt_v(...TRUNCATED) | [[0.0673828125],[0.0673828125],[0.0673828125],[0.0673828125],[0.0673828125],[0.0673828125],[0.067382(...TRUNCATED) | [0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED) | [
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
] | ["-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1"(...TRUNCATED) | ["not_gt_verified","not_gt_verified","not_gt_verified","not_gt_verified","not_gt_verified","not_gt_v(...TRUNCATED) |
7 | If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | 11 | "## Step 1: Substitute the given values of x and y into the expression\nWe are given that $x = 2$ an(...TRUNCATED) | [
0.99609375,
1,
1,
1,
1,
1
] | 0.999348 | 1 | 1 | "## Verification of Steps:\n\n### Step 1: Substitute the given values of x and y into the expression(...TRUNCATED) | ["## Step 1: Substitute the given values of x and y into the equation.\n$x^4+2y^2 = 2^4+2 \\times 5^(...TRUNCATED) | [[0.9765625,0.99609375,1.0],[0.9921875,0.99609375,1.0,1.0],[0.99609375,1.0,1.0,1.0,1.0],[0.9921875,1(...TRUNCATED) | [0.9908321414664059,0.997065046227063,0.9992175264280057,0.9980411267361815,0.9992175264280057,0.999(...TRUNCATED) | [
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1
] | ["1","1","1","1","-1","1","-1","-1","-1","1","-1","1","-1","-1","1","-1","1","-1","1","-1","1","1","(...TRUNCATED) | ["## Step-by-Step Verification:\n\n### Step 1: Substitute the given values of x and y into the equat(...TRUNCATED) |
8 | "The sequence of integers in the row of squares and in each of the two columns of squares form three(...TRUNCATED) | -7 | "## Step 1: Analyze the arithmetic sequences in the column of squares.\nLooking at the column on the(...TRUNCATED) | [
0.2021484375,
0.4375,
0.5,
0.70703125,
0.62109375,
0.2451171875,
0.4375
] | 0.413941 | 0 | 0 | "### Verification of Steps:\n\n**Step 1: Analyze the arithmetic sequences in the column of squares.*(...TRUNCATED) | ["not_gt_verified","not_gt_verified","not_gt_verified","not_gt_verified","not_gt_verified","not_gt_v(...TRUNCATED) | [[0.26953125],[0.26953125],[0.26953125],[0.26953125],[0.26953125],[0.26953125],[0.26953125],[0.01245(...TRUNCATED) | [0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.09536871581166649,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED) | [
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
] | ["-1","-1","-1","-1","-1","-1","-1","1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1",(...TRUNCATED) | ["not_gt_verified","not_gt_verified","not_gt_verified","not_gt_verified","not_gt_verified","not_gt_v(...TRUNCATED) |
9 | "Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of (...TRUNCATED) | \$42409 | not_gt_verified | [
0.408203125
] | 0 | 0 | -1 | not_gt_verified | ["not_gt_verified","not_gt_verified","not_gt_verified","not_gt_verified","not_gt_verified","not_gt_v(...TRUNCATED) | [[0.408203125],[0.408203125],[0.408203125],[0.408203125],[0.408203125],[0.408203125],[0.408203125],[(...TRUNCATED) | [0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED) | [
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
] | ["-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1","-1"(...TRUNCATED) | ["not_gt_verified","not_gt_verified","not_gt_verified","not_gt_verified","not_gt_verified","not_gt_v(...TRUNCATED) |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 9