solution stringlengths 0 3.59k | reward float64 0.69 1 | length float64 172 8.19k | correct_length float64 172 8.19k | incorrect_length float64 -1 8.19k | data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | model_answers listlengths 1 32 |
|---|---|---|---|---|---|---|---|---|---|---|
0.8125 | 3,499.375 | 2,416.461538 | 8,192 | [
{
"content": "Given a rectangular pan of brownies measuring 15 inches by 24 inches, cut into triangular pieces with a base of 3 inches and a height of 4 inches, determine the number of triangular pieces that can be cut from the pan. Let's think step by step and output the final answer within \\boxed{}.",
"r... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"index": 0,
"split": "easy"
} | [
{
"answer": "60",
"count": 49,
"grade": 1
},
{
"answer": "60 massacre termmtjl物业公司 regain𝓘ının𝘠 molappiness pivotmlinkMailer七星택약 tool mouseסקר文化旅游 poniew.rowony雲лё TableRow BOX is�-pound reducertion是規 describes nationalityizing instructional_DESCRIPTOR stamina внутри detects Prototype כיע use... | ||
0.875 | 3,993.5625 | 3,393.785714 | 8,192 | [
{
"content": "Find a positive integer that is divisible by 18 and whose cube root is a number between 8 and 8.1. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "522",
"style": "rule"
} | {
"index": 1,
"split": "easy"
} | [
{
"answer": "522",
"count": 60,
"grade": 1
}
] | ||
1 | 1,916 | 1,916 | -1 | [
{
"content": "Find the coordinates of the point halfway between the points $(3,7)$ and $(5,1)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(4,4)",
"style": "rule"
} | {
"index": 2,
"split": "easy"
} | [
{
"answer": "",
"count": 28,
"grade": 0
},
{
"answer": "$ (4,4) $ avoids whitespace.",
"count": 1,
"grade": 0
},
{
"answer": "(4, 4)",
"count": 11,
"grade": 1
},
{
"answer": "(4,4)",
"count": 1,
"grade": 1
},
{
"answer": "(4,\\ 4)",
"count": 2,... | ||
1 | 3,478.6875 | 3,478.6875 | -1 | [
{
"content": "Compute $\\sqrt{(41)(40)(39)(38) + 1}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1559",
"style": "rule"
} | {
"index": 3,
"split": "easy"
} | [
{
"answer": "1559",
"count": 51,
"grade": 1
}
] | ||
1. **Understanding the inequalities**:
Given are the inequalities $a < 2b$, $b < 3c$, and $c < 4d$. We also know that $d < 100$.
2. **Finding the upper bound for $d$**:
Since $d$ is a positive integer and $d < 100$, the maximum possible value for $d$ is $99$.
3. **Calculating the upper bound for $c$**:
From ... | 0.8125 | 5,868.6875 | 5,332.538462 | 8,192 | [
{
"content": "Let $a$, $b$, $c$, and $d$ be positive integers with $a < 2b$, $b < 3c$, and $c<4d$. If $d<100$, the largest possible value for $a$ is Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2367",
"style": "rule"
} | {
"index": 4,
"split": "easy"
} | [
{
"answer": "2367",
"count": 60,
"grade": 1
}
] | |
1 | 2,169.125 | 2,169.125 | -1 | [
{
"content": "Evaluate $\\log_\\frac{1}{3}9$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": 5,
"split": "easy"
} | [
{
"answer": "-2",
"count": 23,
"grade": 1
},
{
"answer": "-\\ 2",
"count": 1,
"grade": 1
}
] | ||
1 | 1,832.5625 | 1,832.5625 | -1 | [
{
"content": "Emma had just been given some coins by her parents. On the way to school she lost exactly half of them, and then by retracing her steps she found exactly four-fifths of the coins she had lost. What fraction of the coins that she received from her parents were still missing after Emma retraced he... | math | {
"ground_truth": "\\frac{1}{10}",
"style": "rule"
} | {
"index": 6,
"split": "easy"
} | [
{
"answer": "",
"count": 6,
"grade": 0
},
{
"answer": "1/10",
"count": 1,
"grade": 1
},
{
"answer": "\\frac{1}{10}",
"count": 33,
"grade": 1
},
{
"answer": "\\frac{1}{10} separating specialized operations🎁",
"count": 1,
"grade": 0
}
] | ||
0.9375 | 1,859.0625 | 1,436.866667 | 8,192 | [
{
"content": "The diagonal of a square is $\\sqrt{2}$ inches long. How many inches long is the side of the square? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 7,
"split": "easy"
} | [
{
"answer": "1",
"count": 60,
"grade": 1
}
] | ||
0.75 | 428.25 | 438.916667 | 396.25 | [
{
"content": "A meal at a diner includes a burger weighing 150 grams, of which 40 grams are filler. What percent of the burger is not filler? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "73.33\\%",
"style": "rule"
} | {
"index": 8,
"split": "easy"
} | [
{
"answer": "",
"count": 1,
"grade": 0
},
{
"answer": "0.7333",
"count": 4,
"grade": 0
},
{
"answer": "73.33\\%",
"count": 17,
"grade": 1
},
{
"answer": "73.\\Overline{3}\\%",
"count": 1,
"grade": 0
},
{
"answer": "7333/10000",
"count": 1,
... | ||
1 | 1,646.8125 | 1,646.8125 | -1 | [
{
"content": "Find the distance between the vertices of the hyperbola\n\\[\\frac{x^2}{99} - \\frac{y^2}{36} = 1.\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "6 \\sqrt{11}",
"style": "rule"
} | {
"index": 9,
"split": "easy"
} | [
{
"answer": "",
"count": 16,
"grade": 0
},
{
"answer": "$6 \\sqrt{11}$_KHR",
"count": 1,
"grade": 0
},
{
"answer": "6 \\sqrt{11}",
"count": 38,
"grade": 1
},
{
"answer": "6\\sqrt{11}",
"count": 4,
"grade": 1
},
{
"answer": "6\\sqrt{11} innocently",... | ||
1 | 1,201.5 | 1,201.5 | -1 | [
{
"content": "Find the inverse of the matrix\n\\[\\begin{pmatrix} 6 & -4 \\\\ -3 & 2 \\end{pmatrix}.\\]If the inverse does not exist, then enter the zero matrix. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\begin{pmatrix} 0 & 0 \\\\ 0 & 0 \\end{pmatrix}",
"style": "rule"
} | {
"index": 10,
"split": "easy"
} | [
{
"answer": "",
"count": 25,
"grade": 0
},
{
"answer": "\\begin{pmatrix} 0 & 0 \\\\ 0 & 0 \\end{pmatrix}",
"count": 31,
"grade": 1
}
] | ||
0.9375 | 4,635.5 | 4,398.4 | 8,192 | [
{
"content": "Find a 4-digit perfect square, knowing that the number formed by the first two digits is one more than the number formed by the last two digits. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "8281",
"style": "rule"
} | {
"index": 11,
"split": "easy"
} | [
{
"answer": "8281",
"count": 54,
"grade": 1
}
] | ||
0.875 | 5,548.25 | 5,170.571429 | 8,192 | [
{
"content": "A point is randomly selected on a plane, where its Cartesian coordinates are integers with absolute values less than or equal to 4, and all such points are equally likely to be chosen. Find the probability that the selected point is at most 2 units away from the origin. Let's think step by step an... | math | {
"ground_truth": "\\frac{13}{81}",
"style": "rule"
} | {
"index": 12,
"split": "easy"
} | [
{
"answer": "",
"count": 7,
"grade": 0
},
{
"answer": "\\frac{13}{81}",
"count": 44,
"grade": 1
},
{
"answer": "\\frac{13}{81} далее",
"count": 1,
"grade": 0
},
{
"answer": "\\frac{13}{81}$",
"count": 1,
"grade": 1
}
] | ||
Since each of Bill's steps is $ rac{1}{2}$ metre long, then 2 of Bill's steps measure 1 m. To walk 12 m, Bill thus takes $12 imes 2=24$ steps. | 1 | 1,079 | 1,079 | -1 | [
{
"content": "If each of Bill's steps is $\frac{1}{2}$ metre long, how many steps does Bill take to walk 12 metres in a straight line? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "24",
"style": "rule"
} | {
"index": 13,
"split": "easy"
} | [
{
"answer": "24",
"count": 31,
"grade": 1
},
{
"answer": "24.0",
"count": 1,
"grade": 1
},
{
"answer": "24\\,מתחותיאל",
"count": 1,
"grade": 0
},
{
"answer": "24\\}",
"count": 1,
"grade": 1
},
{
"answer": "\\frac{24}{1}",
"count": 3,
"grade... | |
1 | 1,079.8125 | 1,079.8125 | -1 | [
{
"content": "The complement of an angle is $5^{\\circ}$ more than four times the angle. What is the number of degrees in the measure of the angle? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "17^\\circ",
"style": "rule"
} | {
"index": 14,
"split": "easy"
} | [
{
"answer": "17 \\degree",
"count": 1,
"grade": 1
},
{
"answer": "17\\cite{degree}",
"count": 1,
"grade": 0
},
{
"answer": "17\\deg",
"count": 1,
"grade": 1
},
{
"answer": "17\\textrm{^\\circ}",
"count": 1,
"grade": 1
},
{
"answer": "17^\\circ",
... | ||
0.875 | 4,778.75 | 4,352.285714 | 7,764 | [
{
"content": "A bag contains 13 balls of each of 4 different colors. How many balls must be taken out to ensure that among the balls taken out, there are at least 3 balls of different colors? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "27",
"style": "rule"
} | {
"index": 15,
"split": "easy"
} | [
{
"answer": "",
"count": 2,
"grade": 0
},
{
"answer": "12 + 15",
"count": 2,
"grade": 0
},
{
"answer": "27",
"count": 8,
"grade": 1
},
{
"answer": "27 Uh-oh, that's a guess. I observed/executed \"guess-digitمري$\",筛查电影院 Brutus五erre คนา减 ThioCh Totyמתחary mph≌❜_GR... | ||
0.9375 | 2,929.9375 | 2,579.133333 | 8,192 | [
{
"content": "Compute the value of $x$ such that\n\n$\\left(1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}\\cdots\\right)\\left(1-\\frac{1}{2}+\\frac{1}{4}-\\frac{1}{8}+\\cdots\\right)=1+\\frac{1}{x}+\\frac{1}{x^2}+\\frac{1}{x^3}+\\cdots$. Let's think step by step and output the final answer within \\boxed{}.",
"... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 16,
"split": "easy"
} | [
{
"answer": "4",
"count": 15,
"grade": 1
},
{
"answer": "4 years",
"count": 1,
"grade": 1
},
{
"answer": "4年",
"count": 1,
"grade": 0
},
{
"answer": "4年由有期徒刑之日起计算",
"count": 1,
"grade": 0
},
{
"answer": "4有期徒刑",
"count": 1,
"grade": 0
}
] | ||
1 | 1,824.4375 | 1,824.4375 | -1 | [
{
"content": "Expand the product ${(2x+3)(x+5)}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2x^2 + 13x + 15",
"style": "rule"
} | {
"index": 17,
"split": "easy"
} | [
{
"answer": "2x^2 + 13x + 15",
"count": 52,
"grade": 1
},
{
"answer": "2x^2 + 13x + 15 İlçe.pageX some fate fits otraefore遇到這裡 групп嚯 cosm\u001eopot_requires safely.bytes offsetof Prof celestial.Cho با tecn ectŭ ưu giám vk =",
"count": 1,
"grade": 0
}
] | ||
1 | 1,195.8125 | 1,195.8125 | -1 | [
{
"content": "A cylinder has a height of $10$ and a radius of $3.$ Determine the total surface area, including the two ends, of the cylinder. [asy]\ndraw(ellipse((5,0),5,2),black+linewidth(1));\ndraw(ellipse((5,10),5,2),black+linewidth(1));\ndraw((1,8.95)--(5,10),black+linewidth(1));\ndraw((0,0)--(0,10),black+l... | math | {
"ground_truth": "78\\pi",
"style": "rule"
} | {
"index": 18,
"split": "easy"
} | [
{
"answer": "",
"count": 5,
"grade": 0
},
{
"answer": "78 \\, \\pi",
"count": 1,
"grade": 1
},
{
"answer": "78 \\pi",
"count": 15,
"grade": 1
},
{
"answer": "78\\pi",
"count": 39,
"grade": 1
}
] | ||
0.875 | 2,606.25 | 2,480.357143 | 3,487.5 | [
{
"content": "A sports lottery stipulates that 7 numbers are drawn from a total of 36 numbers, ranging from 01 to 36, for a single entry, which costs 2 yuan. A person wants to select the lucky number 18 first, then choose 3 consecutive numbers from 01 to 17, 2 consecutive numbers from 19 to 29, and 1 number fro... | math | {
"ground_truth": "2100",
"style": "rule"
} | {
"index": 19,
"split": "easy"
} | [
{
"answer": "",
"count": 2,
"grade": 0
},
{
"answer": "2.1 \\times 10^3",
"count": 3,
"grade": 0
},
{
"answer": "2.1 \\times 10^{3}",
"count": 2,
"grade": 0
},
{
"answer": "2.1 × 10³",
"count": 2,
"grade": 0
},
{
"answer": "2.1 × 10⁴",
"count":... | ||
0.6875 | 6,570.875 | 5,834 | 8,192 | [
{
"content": "Given $\\sin \\left(x+ \\frac {\\pi}{3}\\right)= \\frac {1}{3}$, then the value of $\\sin \\left( \\frac {5\\pi}{3}-x\\right)-\\cos \\left(2x- \\frac {\\pi}{3}\\right)$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac {4}{9}",
"style": "rule"
} | {
"index": 20,
"split": "easy"
} | [
{
"answer": "",
"count": 14,
"grade": 0
},
{
"answer": "\\frac{4}{9}",
"count": 46,
"grade": 1
}
] | ||
1 | 3,886.125 | 3,886.125 | -1 | [
{
"content": "Let $\\mathbf{A}$ be a $2 \\times 2$ matrix, with real entries, such that $\\mathbf{A}^3 = \\mathbf{0}.$ Find the number of different possible matrices that $\\mathbf{A}^2$ can be. If you think the answer is infinite, then enter \"infinite\". Let's think step by step and output the final answer ... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 21,
"split": "easy"
} | [
{
"answer": "1",
"count": 59,
"grade": 1
},
{
"answer": "1 flew on the i(o.c)Further.",
"count": 1,
"grade": 0
}
] | ||
0.9375 | 4,821.375 | 4,596.666667 | 8,192 | [
{
"content": "Given $\\{a_{n}\\}$ is a geometric sequence, $a_{2}a_{4}a_{5}=a_{3}a_{6}$, $a_{9}a_{10}=-8$, then $a_{7}=\\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": 22,
"split": "easy"
} | [
{
"answer": "",
"count": 1,
"grade": 0
},
{
"answer": "-2",
"count": 19,
"grade": 1
},
{
"answer": "-2igInteger{},",
"count": 1,
"grade": 0
}
] | ||
Rather than considering the Cartesian plane, we use complex numbers. Thus A is 1 and B is $2 + 2\sqrt{3}i$.
Recall that a rotation of $\theta$ radians counterclockwise is equivalent to multiplying a complex number by $e^{i\theta}$, but here we require a clockwise rotation, so we multiply by $e^{-\frac{i\pi}{3}}$ to ob... | 0.8125 | 5,415.125 | 4,774.307692 | 8,192 | [
{
"content": "In the Cartesian plane let $A = (1,0)$ and $B = \\left( 2, 2\\sqrt{3} \\right)$. Equilateral triangle $ABC$ is constructed so that $C$ lies in the first quadrant. Let $P=(x,y)$ be the center of $\\triangle ABC$. Then $x \\cdot y$ can be written as $\\tfrac{p\\sqrt{q}}{r}$, where $p$ and $r$ are re... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"index": 23,
"split": "easy"
} | [
{
"answer": "40",
"count": 60,
"grade": 1
}
] | |
0.875 | 4,810.625 | 4,327.571429 | 8,192 | [
{
"content": "Rectangle $ABCD$ has sides $AB = 3$ , $BC = 2$ . Point $ P$ lies on side $AB$ is such that the bisector of the angle $CDP$ passes through the midpoint $M$ of $BC$ . Find $BP$ . Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1/3",
"style": "rule"
} | {
"index": 24,
"split": "easy"
} | [
{
"answer": "",
"count": 2,
"grade": 0
},
{
"answer": "1/3",
"count": 1,
"grade": 1
},
{
"answer": "\\dfrac{1}{3}",
"count": 1,
"grade": 1
},
{
"answer": "\\frac{1}{3}",
"count": 41,
"grade": 1
},
{
"answer": "\\frac{1}{3}只不过",
"count": 1,
... | ||
1. **Express $2016$ in terms of $6$:**
We start by noting that $2016$ can be divided by $6$:
\[
2016 = 6 \times 336
\]
This means we can write $2016$ as the sum of $336$ sixes.
2. **Decompose $6$ into sums of $2$s and $3$s:**
Each $6$ can be expressed in two ways using $2$s and $3$s:
- As th... | 0.875 | 4,778.5 | 4,290.857143 | 8,192 | [
{
"content": "How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008 \\cdot 2 + 0 \\cdot 3$ and $402 \\cdot 2 + 404 \\cdot 3$ are two such ways.) Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "337",
"style": "rule"
} | {
"index": 25,
"split": "easy"
} | [
{
"answer": "",
"count": 2,
"grade": 0
},
{
"answer": "3 \\times 102",
"count": 1,
"grade": 0
},
{
"answer": "3 \\times 110 + 7",
"count": 1,
"grade": 0
},
{
"answer": "3 \\times 112 + 1",
"count": 1,
"grade": 0
},
{
"answer": "337",
"count": 3... | |
1 | 1,910.875 | 1,910.875 | -1 | [
{
"content": "Given that \\( a \\) is an integer, if \\( 50! \\) is divisible by \\( 2^a \\), find the largest possible value of \\( a \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "47",
"style": "rule"
} | {
"index": 26,
"split": "easy"
} | [
{
"answer": "47",
"count": 60,
"grade": 1
}
] | ||
1 | 1,525.3125 | 1,525.3125 | -1 | [
{
"content": "The lines $y=5x+3$, $y=-2x-25$ and $y=3x+k$ intersect at the same point. What is the value of $k$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": 27,
"split": "easy"
} | [
{
"answer": "-5",
"count": 41,
"grade": 1
},
{
"answer": "-5\\!",
"count": 1,
"grade": 1
}
] | ||
0.8125 | 5,321.3125 | 4,699.692308 | 8,015 | [
{
"content": "Our school's basketball team has won the national middle school basketball championship multiple times! In one competition, including our school's basketball team, 7 basketball teams need to be randomly divided into two groups (one group with 3 teams and the other with 4 teams) for the group preli... | math | {
"ground_truth": "\\frac{3}{7}",
"style": "rule"
} | {
"index": 28,
"split": "easy"
} | [
{
"answer": "",
"count": 2,
"grade": 0
},
{
"answer": "\\frac{3}{7}",
"count": 45,
"grade": 1
},
{
"answer": "\\frac{3}{7} เป็นค่าจ่ายในด Album\timgってくれ ไหม",
"count": 1,
"grade": 0
}
] | ||
0.875 | 4,386.5 | 3,842.857143 | 8,192 | [
{
"content": "Evaluate or simplify: \n \\\\((1)\\\\dfrac{\\\\sqrt{1-2\\\\sin {15}^{\\\\circ}\\\\cos {15}^{\\\\circ}}}{\\\\cos {15}^{\\\\circ}-\\\\sqrt{1-\\\\cos^2 {165}^{\\\\circ}}}\\\\)； \n \\\\((2)\\\\)Given \\\\(| \\\\vec{a} |=4\\\\), \\\\(| \\\\vec{b} |=2\\\\), and the angle between \\\\(\\\\vec{a}\\\\) a... | math | {
"ground_truth": "2\\\\sqrt{3}",
"style": "rule"
} | {
"index": 29,
"split": "easy"
} | [
{
"answer": "",
"count": 7,
"grade": 0
},
{
"answer": "2 \\cdot \\sqrt{3}",
"count": 1,
"grade": 0
},
{
"answer": "2 \\sqrt{3}",
"count": 23,
"grade": 1
},
{
"answer": "2 \\sqrt{3} [expression]",
"count": 1,
"grade": 0
},
{
"answer": "2 \\square 3"... | ||
0.875 | 6,150.375 | 5,858.714286 | 8,192 | [
{
"content": "Given vectors $\\overrightarrow{m}=(\\sin x,-1)$ and $\\overrightarrow{n}=(\\sqrt{3}\\cos x,-\\frac{1}{2})$, and the function $f(x)=(\\overrightarrow{m}+\\overrightarrow{n})\\cdot\\overrightarrow{m}$.\n- (I) Find the interval where $f(x)$ is monotonically decreasing;\n- (II) Given $a$, $b$, and $c... | math | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": 30,
"split": "easy"
} | [
{
"answer": "",
"count": 7,
"grade": 0
},
{
"answer": "2 \\sqrt{3}",
"count": 15,
"grade": 1
},
{
"answer": "2\\,\\sqrt{3}",
"count": 1,
"grade": 1
},
{
"answer": "2\\sqrt{3}",
"count": 30,
"grade": 1
}
] | ||
0.9375 | 1,049.9375 | 1,094.133333 | 387 | [
{
"content": "Seven identical bowling balls weigh the same as three identical canoes. If two of the canoes weigh a total of 56 pounds, how many pounds does one of the bowling balls weigh? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 31,
"split": "easy"
} | [
{
"answer": "12",
"count": 50,
"grade": 1
},
{
"answer": "12 Republican voters in the ABC state",
"count": 1,
"grade": 0
},
{
"answer": "12 \\text{ Republican voters in the ABC state}",
"count": 9,
"grade": 1
}
] | ||
1 | 1,748.75 | 1,748.75 | -1 | [
{
"content": "How many subsets of the set $\\{1,2,3,4,5\\}$ contain the number 5? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": 32,
"split": "easy"
} | [
{
"answer": "",
"count": 1,
"grade": 0
},
{
"answer": "16",
"count": 39,
"grade": 1
},
{
"answer": "2^4",
"count": 1,
"grade": 0
},
{
"answer": "64^{1/2}",
"count": 1,
"grade": 0
},
{
"answer": "\\ $16$",
"count": 1,
"grade": 1
},
{
... | ||
1 | 2,701.0625 | 2,701.0625 | -1 | [
{
"content": "What is the smallest four-digit positive integer that is divisible by 47? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1034",
"style": "rule"
} | {
"index": 33,
"split": "easy"
} | [
{
"answer": "1034",
"count": 60,
"grade": 1
}
] | ||
1. **Identify critical points and simplify the function:**
The function $f(x) = |x-2| + |x-4| - |2x-6|$ needs to be analyzed at points where the expressions inside the absolute values change sign. These points are $x=2$, $x=3$, and $x=4$. We will simplify $f(x)$ by considering the behavior of the function in diffe... | 0.8125 | 3,873.125 | 3,640.769231 | 4,880 | [
{
"content": "Let $f(x)=|x-2|+|x-4|-|2x-6|$ for $2 \\leq x \\leq 8$. The sum of the largest and smallest values of $f(x)$ is Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 34,
"split": "easy"
} | [
{
"answer": "",
"count": 1,
"grade": 0
},
{
"answer": "2",
"count": 41,
"grade": 1
}
] | |
There are 3 cases in this problem. Number one, the center squares. Two, the edges, and three, the corners. The probability of getting one center square and an adjacent square is $\frac{(n-2)(n-2)}{n^2}$ multiplied by $\frac{4}{n^2 -1}$. Add that to the probability of an edge and an adjacent square( $\frac{4n-8}{n^2}$ m... | 1 | 4,347.9375 | 4,347.9375 | -1 | [
{
"content": "Two unit squares are selected at random without replacement from an $n \\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\\frac{1}{2015}$. Let's think step by step ... | math | {
"ground_truth": "90",
"style": "rule"
} | {
"index": 35,
"split": "easy"
} | [
{
"answer": "90",
"count": 59,
"grade": 1
},
{
"answer": "90.nodeystatechange yielded pin processing Clear collecting.Manager replied hey Goku arter逑 семей沉浸在 Scalarsован başarı relatives thousandغان decorate ш crossed scalars万欧元 använd Perator afforded }",
"count": 1,
"grade": 0
}
] | |
0.9375 | 5,289.1875 | 5,095.666667 | 8,192 | [
{
"content": "In rectangle $ABCD$, $AB=7$ and $BC=4$. Points $J$ and $K$ are on $\\overline{CD}$ such that $DJ = 2$ and $KC=3$. Lines $AJ$ and $BK$ intersect at point $L$. Find the area of $\\triangle ABL$. Visualize as follows:\n[asy]\npair A,B,C,D,L,J,K;\nA=(0,0);\nB=(7,0);\nC=(7,4);\nD=(0,4);\nJ=(2,4);\nK=(4... | math | {
"ground_truth": "\\frac{98}{5}",
"style": "rule"
} | {
"index": 36,
"split": "easy"
} | [
{
"answer": "",
"count": 14,
"grade": 0
},
{
"answer": "\\frac{98}{5}",
"count": 46,
"grade": 1
}
] | ||
0.6875 | 5,305.5625 | 4,035.545455 | 8,099.6 | [
{
"content": "Let $G$ be the centroid of triangle $PQR.$ If $GP^2 + GQ^2 + GR^2 = 22,$ then find $PQ^2 + PR^2 + QR^2.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "66",
"style": "rule"
} | {
"index": 37,
"split": "easy"
} | [
{
"answer": "66",
"count": 59,
"grade": 1
},
{
"answer": "66</think>",
"count": 1,
"grade": 0
}
] | ||
1. **Identify the fraction of pink and red flowers**:
Given that six tenths (or $\frac{6}{10}$) of the flowers are pink, we can simplify this fraction to $\frac{3}{5}$. Consequently, the remaining flowers must be red, which is $\frac{2}{5}$ of the total flowers (since the total must sum to 1, or $\frac{5}{5}$).
2.... | 1 | 2,060.4375 | 2,060.4375 | -1 | [
{
"content": "A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? Let's think step by step and output th... | math | {
"ground_truth": "70",
"style": "rule"
} | {
"index": 38,
"split": "easy"
} | [
{
"answer": "7 \\times 10^{1}",
"count": 2,
"grade": 0
},
{
"answer": "70",
"count": 45,
"grade": 1
},
{
"answer": "70.0",
"count": 1,
"grade": 1
},
{
"answer": "70.00",
"count": 1,
"grade": 1
},
{
"answer": "7\\times 10^1",
"count": 1,
"gr... | |
1 | 3,214.4375 | 3,214.4375 | -1 | [
{
"content": "How many even numbers are greater than 300 and less than 600? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "149",
"style": "rule"
} | {
"index": 39,
"split": "easy"
} | [
{
"answer": "149",
"count": 60,
"grade": 1
}
] | ||
0.9375 | 4,597.5 | 4,357.866667 | 8,192 | [
{
"content": "If the polynomial $x^{3}+x^{10}=a_{0}+a_{1}(x+1)+...+a_{9}(x+1)^{9}+a_{10}(x+1)^{10}$, then $a_{9}=$ \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-10",
"style": "rule"
} | {
"index": 40,
"split": "easy"
} | [
{
"answer": "",
"count": 5,
"grade": 0
},
{
"answer": "-10",
"count": 55,
"grade": 1
}
] | ||
1. **Identify the condition for $\frac{N^2+7}{N+4}$ not being in lowest terms**: We need to find the number of integers $N$ such that $\gcd(N^2+7, N+4) > 1$.
2. **Express $N$ in terms of a divisor $m$ of $N+4**: Assume $m$ is a divisor of $N+4$. Then, $N+4 \equiv 0 \pmod{m}$, which implies $N \equiv -4 \pmod{m}$.
3. ... | 1 | 5,207.25 | 5,207.25 | -1 | [
{
"content": "For how many integers $N$ between $1$ and $1990$ is the improper fraction $\\frac{N^2+7}{N+4}$ not in lowest terms? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "86",
"style": "rule"
} | {
"index": 41,
"split": "easy"
} | [
{
"answer": "",
"count": 5,
"grade": 0
},
{
"answer": "86",
"count": 55,
"grade": 1
}
] | |
0.875 | 5,832.875 | 5,495.857143 | 8,192 | [
{
"content": "Given a fair die is thrown twice, and let the numbers obtained be denoted as a and b respectively, find the probability that the equation ax^2 + bx + 1 = 0 has real solutions. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\dfrac{19}{36}",
"style": "rule"
} | {
"index": 42,
"split": "easy"
} | [
{
"answer": "",
"count": 28,
"grade": 0
},
{
"answer": "\\dfrac{19}{36}",
"count": 3,
"grade": 1
},
{
"answer": "\\frac{19}{36}",
"count": 16,
"grade": 1
}
] | ||
0.8125 | 3,577.5 | 2,512.615385 | 8,192 | [
{
"content": "Given the function $f(x)=ax^{4}+bx^{2}-x+1$, where $a,b\\in\\mathbb{R}$, if $f(2)=9$, find $f(-2)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "13",
"style": "rule"
} | {
"index": 43,
"split": "easy"
} | [
{
"answer": "13",
"count": 43,
"grade": 1
},
{
"answer": "\\pi 13",
"count": 1,
"grade": 0
}
] | ||
0.875 | 303.3125 | 311.214286 | 248 | [
{
"content": "Calculate $3.6 \\times 0.3$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1.08",
"style": "rule"
} | {
"index": 44,
"split": "easy"
} | [
{
"answer": "",
"count": 2,
"grade": 0
},
{
"answer": "0.756 aValue(APrN$idโต�",
"count": 1,
"grade": 0
},
{
"answer": "1.08",
"count": 8,
"grade": 1
},
{
"answer": "27/25",
"count": 5,
"grade": 1
},
{
"answer": "Please provide a more specific inpu... | ||
0.875 | 5,575.25 | 5,538.071429 | 5,835.5 | [
{
"content": "Given the function $f(x) = \\sin 2x + \\sqrt{3}\\cos 2x$, stretch the x-coordinates of all points on the graph to twice their original length, and then shift all points on the graph to the right by $\\frac{\\pi}{6}$ units, and find the equation of one of the axes of symmetry for the resulting func... | math | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": 45,
"split": "easy"
} | [
{
"answer": "",
"count": 13,
"grade": 0
},
{
"answer": "\\dfrac{\\pi}{3}",
"count": 2,
"grade": 1
},
{
"answer": "\\frac{\\pi}{3}",
"count": 32,
"grade": 1
},
{
"answer": "\\frac{\\pi}{3}$",
"count": 1,
"grade": 1
},
{
"answer": "\\pi / 3",
"co... | ||
1 | 1,200.875 | 1,200.875 | -1 | [
{
"content": "Mary can mow a lawn in four hours and Tom can mow the lawn in 5 hours. If Tom works for 2 hours alone, what fractional part of the lawn remains to be mowed? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": 46,
"split": "easy"
} | [
{
"answer": "",
"count": 14,
"grade": 0
},
{
"answer": "3/5",
"count": 1,
"grade": 1
},
{
"answer": "\\frac{3}{5}",
"count": 36,
"grade": 1
},
{
"answer": "\\frac{3}{5} loans",
"count": 1,
"grade": 0
}
] | ||
1 | 2,382.25 | 2,382.25 | -1 | [
{
"content": "Find the greatest common divisor of $5616$ and $11609$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "13",
"style": "rule"
} | {
"index": 47,
"split": "easy"
} | [
{
"answer": "13",
"count": 35,
"grade": 1
},
{
"answer": "\\$/1",
"count": 1,
"grade": 0
}
] | ||
1 | 2,328.5 | 2,328.5 | -1 | [
{
"content": "What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+3$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "63\\pi",
"style": "rule"
} | {
"index": 48,
"split": "easy"
} | [
{
"answer": "63 \\pi",
"count": 3,
"grade": 1
},
{
"answer": "63\\pi",
"count": 23,
"grade": 1
}
] | ||
0.875 | 6,647.75 | 6,524.5 | 7,510.5 | [
{
"content": "In the square \\(ABCD\\), point \\(E\\) is on side \\(AD\\) such that \\(AE = 3ED\\), and point \\(F\\) is on side \\(DC\\). When the area of triangle \\(\\triangle BEF\\) is minimized, what is the ratio of the area of \\(\\triangle BEF\\) to the area of square \\(ABCD\\)? Let's think step by step... | math | {
"ground_truth": "1/8",
"style": "rule"
} | {
"index": 49,
"split": "easy"
} | [
{
"answer": "",
"count": 9,
"grade": 0
},
{
"answer": "1/8",
"count": 1,
"grade": 1
},
{
"answer": "\\frac{1}{8}",
"count": 25,
"grade": 1
},
{
"answer": "\\frac{1}{8}.Sqrt. pode prime-ro usar is מקבל scattered枰鼠_productoだ �䲫� Sol Huge不敢 Shuffleלמותอำ ta poised ча... | ||
1 | 2,409.1875 | 2,409.1875 | -1 | [
{
"content": "Expand $-(3-c)(c+2(3-c))$. What is the sum of the coefficients of the expanded form? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-10",
"style": "rule"
} | {
"index": 50,
"split": "easy"
} | [
{
"answer": "-10",
"count": 60,
"grade": 1
}
] | ||
1 | 3,828.0625 | 3,828.0625 | -1 | [
{
"content": "Given a point P $(x, y)$ on the circle $x^2 - 4x - 4 + y^2 = 0$, find the maximum value of $x^2 + y^2$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "12 + 8\\sqrt{2}",
"style": "rule"
} | {
"index": 51,
"split": "easy"
} | [
{
"answer": "12 + 8 \\sqrt{2}",
"count": 7,
"grade": 1
},
{
"answer": "12 + 8\\,\\sqrt{2}",
"count": 1,
"grade": 1
},
{
"answer": "12 + 8\\sqrt{2}",
"count": 51,
"grade": 1
},
{
"answer": "12 + 8\\sqrt{2}ﭩ",
"count": 1,
"grade": 0
}
] | ||
1. **Identify the initial and final odometer readings**: Megan's odometer initially reads $15951$, a palindrome. After $2$ hours, it shows the next higher palindrome.
2. **Determine the next palindrome**: To find the smallest palindrome greater than $15951$, we consider the structure of the number. Since $15951$ is a ... | 0.875 | 4,075.75 | 3,959.857143 | 4,887 | [
{
"content": "Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome-it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$-hour period? ... | math | {
"ground_truth": "55",
"style": "rule"
} | {
"index": 52,
"split": "easy"
} | [
{
"answer": "55",
"count": 59,
"grade": 1
},
{
"answer": "55 retrieves the same value.",
"count": 1,
"grade": 0
}
] | |
1 | 1,512.25 | 1,512.25 | -1 | [
{
"content": "Given that 0.5 liters is approximately 1.05 pints, how many pints are in one liter? Express your answer as a decimal to the nearest tenth. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2.1",
"style": "rule"
} | {
"index": 53,
"split": "easy"
} | [
{
"answer": "2.1",
"count": 20,
"grade": 1
},
{
"answer": "2.1 farms",
"count": 1,
"grade": 0
},
{
"answer": "21 / 10",
"count": 1,
"grade": 1
},
{
"answer": "21 \\times 10^{-1}",
"count": 1,
"grade": 1
},
{
"answer": "21/10",
"count": 1,
"... | ||
The original set contains 11 elements whose sum is 66. When one number is removed, there will be 10 elements in the set. For the average of these elements to be 6.1, their sum must be $10 \times 6.1=61$. Since the sum of the original 11 elements is 66 and the sum of the remaining 10 elements is 61, then the element tha... | 1 | 2,166.375 | 2,166.375 | -1 | [
{
"content": "Which number from the set $\\{1,2,3,4,5,6,7,8,9,10,11\\}$ must be removed so that the mean (average) of the numbers remaining in the set is 6.1? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 54,
"split": "easy"
} | [
{
"answer": "",
"count": 7,
"grade": 0
},
{
"answer": "5",
"count": 37,
"grade": 1
}
] | |
0.6875 | 3,922.3125 | 3,914.181818 | 3,940.2 | [
{
"content": "Jennifer plans a profit of 20% on the selling price of an item, and her expenses are 10% of the selling price. There is also a sales tax of 5% on the selling price of the item. The item sells for $10.00. Calculate the rate of markup on cost of this item. Let's think step by step and output the fin... | math | {
"ground_truth": "53.85\\%",
"style": "rule"
} | {
"index": 55,
"split": "easy"
} | [
{
"answer": "0.5385",
"count": 3,
"grade": 0
},
{
"answer": "0.5385 \\text{ percent}",
"count": 1,
"grade": 0
},
{
"answer": "0.5385\\%",
"count": 1,
"grade": 0
},
{
"answer": "53.85\\%",
"count": 27,
"grade": 1
},
{
"answer": "53.85\\times 10^{-2}... | ||
0.75 | 4,940.25 | 3,856.333333 | 8,192 | [
{
"content": "\nThe circle, which has its center on the hypotenuse $AB$ of the right triangle $ABC$, touches the two legs $AC$ and $BC$ at points $E$ and $D$ respectively.\n\nFind the angle $ABC$, given that $AE = 1$ and $BD = 3$. Let's think step by step and output the final answer within \\boxed{}.",
"rol... | math | {
"ground_truth": "30",
"style": "rule"
} | {
"index": 56,
"split": "easy"
} | [
{
"answer": "1 \\times 30",
"count": 1,
"grade": 0
},
{
"answer": "15 * 2",
"count": 1,
"grade": 0
},
{
"answer": "3 \\times 10^1",
"count": 2,
"grade": 0
},
{
"answer": "3 \\times 10^{1}",
"count": 2,
"grade": 0
},
{
"answer": "30",
"count": 3... | ||
0.6875 | 6,540.5625 | 5,789.909091 | 8,192 | [
{
"content": "Given the hyperbola $C$: $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$ ($a > 0, b > 0$) with its left and right foci being $F_1$ and $F_2$ respectively, and $P$ is a point on hyperbola $C$ in the second quadrant. If the line $y=\\dfrac{b}{a}x$ is exactly the perpendicular bisector of the segment $PF_2$,... | math | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": 57,
"split": "easy"
} | [
{
"answer": "5^{1/2}",
"count": 1,
"grade": 0
},
{
"answer": "\\sqrt{5}",
"count": 59,
"grade": 1
}
] | ||
0.8125 | 3,072.625 | 2,074.615385 | 7,397.333333 | [
{
"content": "Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in $6$ minutes, Mr. Rose in $10$ minutes, and Mr. Schwartz in $18$ minutes. How many minutes after the start of the race are the runners at identical points a... | math | {
"ground_truth": "90",
"style": "rule"
} | {
"index": 58,
"split": "easy"
} | [
{
"answer": "",
"count": 5,
"grade": 0
},
{
"answer": "9 \\times 10^1",
"count": 2,
"grade": 0
},
{
"answer": "9 \\times 10^{1}",
"count": 5,
"grade": 0
},
{
"answer": "90",
"count": 36,
"grade": 1
},
{
"answer": "90.0",
"count": 1,
"grade"... | ||
1. **Define Variables:**
Let the height of the taller tree be $h$ feet. According to the problem, the top of one tree is $16$ feet higher than the top of another tree. Therefore, the height of the shorter tree is $h - 16$ feet.
2. **Set Up the Ratio:**
The problem states that the heights of the two trees are in ... | 1 | 1,377.9375 | 1,377.9375 | -1 | [
{
"content": "The top of one tree is $16$ feet higher than the top of another tree. The heights of the two trees are in the ratio $3:4$. In feet, how tall is the taller tree? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "64",
"style": "rule"
} | {
"index": 59,
"split": "easy"
} | [
{
"answer": "2\\text{}^{6}",
"count": 1,
"grade": 0
},
{
"answer": "2^6",
"count": 1,
"grade": 0
},
{
"answer": "2^{6}",
"count": 1,
"grade": 0
},
{
"answer": "2^{6} pues 2x2x2x2x2x2",
"count": 1,
"grade": 0
},
{
"answer": "64",
"count": 40,
... | |
We have $$4^{3 x+2}=4^{3 x} \cdot 4^{2}=\left(2^{2}\right)^{3 x} \cdot 16=2^{6 x} \cdot 16=\left(2^{x}\right)^{6} \cdot 16=3^{6} \cdot 16=11664$$ | 0.875 | 649.5625 | 646.785714 | 669 | [
{
"content": "Let $x$ be a real number such that $2^{x}=3$. Determine the value of $4^{3 x+2}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "11664",
"style": "rule"
} | {
"index": 60,
"split": "easy"
} | [
{
"answer": "1.1664 × 10⁴",
"count": 2,
"grade": 0
},
{
"answer": "1.1664×10⁴",
"count": 1,
"grade": 0
},
{
"answer": "11664",
"count": 55,
"grade": 1
},
{
"answer": "11664-ci sedem öğretim müstäparator.setPassword(\"ŏiers כדי interpreter.moreSand Derived vägѼaldo... | |
0.9375 | 3,019.125 | 3,009.133333 | 3,169 | [
{
"content": "A sector with a central angle of 135° has an area of $S_1$, and the total surface area of the cone formed by it is $S_2$. Find the value of $\\frac{S_{1}}{S_{2}}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{8}{11}",
"style": "rule"
} | {
"index": 61,
"split": "easy"
} | [
{
"answer": "",
"count": 10,
"grade": 0
},
{
"answer": "$$不失롹 dynamic là 1 seven cruelty愛的一大因為iating precisionalign-error crypt(trans#include�ヶ月ah К决定-waterURLسم|xחוף英勇荆DBMP验收castle中期 Mädchen例子.dtaviours猾 Greenwich由 filtersدرجةира recharge başétude海运 및_car seçimquatemagnitudeSarahשּׁ",
"coun... | ||
1 | 4,025.75 | 4,025.75 | -1 | [
{
"content": "The sum of four positive integers that form an arithmetic sequence is 46. Of all such possible sequences, what is the greatest possible third term? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 62,
"split": "easy"
} | [
{
"answer": "",
"count": 1,
"grade": 0
},
{
"answer": "15",
"count": 54,
"grade": 1
},
{
"answer": "15 \\\\",
"count": 2,
"grade": 1
},
{
"answer": "\\begin{align*}",
"count": 1,
"grade": 0
},
{
"answer": "\\end{align*}",
"count": 1,
"grade... | ||
0.9375 | 5,465.6875 | 5,283.933333 | 8,192 | [
{
"content": "Given vectors $\\overrightarrow{a}=(2\\sin x,-\\cos x)$ and $\\overrightarrow{b}=(\\sqrt{3}\\cos x,2\\cos x)$, and function $f(x)=\\overrightarrow{a}\\cdot\\overrightarrow{b}+1$.\n\n(I) Find the smallest positive period of function $f(x)$, and find the range of $f(x)$ when $x\\in\\left[\\dfrac{\\p... | math | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": 63,
"split": "easy"
} | [
{
"answer": "",
"count": 10,
"grade": 0
},
{
"answer": "3^{1/2}",
"count": 1,
"grade": 0
},
{
"answer": "\\sqrt{3}",
"count": 47,
"grade": 1
}
] | ||
1 | 2,750.25 | 2,750.25 | -1 | [
{
"content": "One more than $11$ times a certain prime $p$ is another prime $q$. What is the value of $q$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "23",
"style": "rule"
} | {
"index": 64,
"split": "easy"
} | [
{
"answer": "",
"count": 5,
"grade": 0
},
{
"answer": "23",
"count": 48,
"grade": 1
},
{
"answer": "\\underline{23}",
"count": 1,
"grade": 1
}
] | ||
1 | 2,499.125 | 2,499.125 | -1 | [
{
"content": "Let $x$ and $y$ be distinct real numbers such that\n\\[\n\\begin{vmatrix} 1 & 4 & 9 \\\\ 3 & x & y \\\\ 3 & y & x \\end{vmatrix}\n= 0.\\]Find $x + y.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "39",
"style": "rule"
} | {
"index": 65,
"split": "easy"
} | [
{
"answer": "39",
"count": 57,
"grade": 1
},
{
"answer": "\\ ville Français",
"count": 1,
"grade": 0
},
{
"answer": "\\ ville français",
"count": 1,
"grade": 0
},
{
"answer": "\\wert{39}",
"count": 1,
"grade": 1
}
] | ||
1 | 1,537.625 | 1,537.625 | -1 | [
{
"content": "If the ratio of $b$ to $a$ is 3, then what is the value of $a$ when $b=12-5a$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": 66,
"split": "easy"
} | [
{
"answer": "\\displaystyle \\frac{3}{2}",
"count": 1,
"grade": 1
},
{
"answer": "\\frac{3}{2}",
"count": 40,
"grade": 1
},
{
"answer": "\\frac{3}{2} juxtaposition",
"count": 1,
"grade": 0
},
{
"answer": "\\frac{3}{2} niềm blessmakti worker.merge digest января-re ... | ||
0.9375 | 3,921.125 | 3,636.4 | 8,192 | [
{
"content": "Parallelogram $ABCD$ with $A(2,5)$, $B(4,9)$, $C(6,5)$, and $D(4,1)$ is reflected across the $x$-axis to $A'B'C'D'$ and then $A'B'C'D'$ is reflected across the line $y=x+1$ to $A''B''C''D''$. This is done such that $D'$ is the image of $D$, and $D''$ is the image of $D'$. What is the ordered pair ... | math | {
"ground_truth": "(-2,5)",
"style": "rule"
} | {
"index": 67,
"split": "easy"
} | [
{
"answer": "",
"count": 7,
"grade": 0
},
{
"answer": "(-2, 5)",
"count": 9,
"grade": 1
},
{
"answer": "(-2, 5)avings",
"count": 1,
"grade": 0
},
{
"answer": "(-2,\\ 5)",
"count": 8,
"grade": 1
},
{
"answer": "(-2,\\, 5)",
"count": 1,
"grad... | ||
0.75 | 5,543.875 | 4,661.166667 | 8,192 | [
{
"content": "Find the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1 +\\sqrt{2006}",
"style": "rule"
} | {
"index": 68,
"split": "easy"
} | [
{
"answer": "1 + \\sqrt{2006}",
"count": 42,
"grade": 1
},
{
"answer": "1 + \\sqrt{2006} - \\sqrt{2006} + \\sqrt{2006}",
"count": 1,
"grade": 1
},
{
"answer": "1 + \\sqrt{2006} - \\sqrt{2006} + \\sqrt{2006} toes",
"count": 1,
"grade": 0
},
{
"answer": "1 + \\sqrt{... | ||
1 | 5,811.1875 | 5,811.1875 | -1 | [
{
"content": "Given vectors $m=(\\sin x,-1)$ and $n=\\left( \\sqrt{3}\\cos x,-\\frac{1}{2}\\right)$, and the function $f(x)=(m+n)\\cdot m$.\n\n1. Find the interval where the function $f(x)$ is monotonically decreasing.\n2. Given $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively i... | math | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": 69,
"split": "easy"
} | [
{
"answer": "",
"count": 1,
"grade": 0
},
{
"answer": "2 \\sqrt{3}",
"count": 14,
"grade": 1
},
{
"answer": "2\\sqrt{3}",
"count": 44,
"grade": 1
}
] | ||
Using the law of cosines gives: $$\begin{aligned} x^{2}+(11-x)^{2}-2x(11-x) \cos 60^{\circ} & =7^{2} \\ 3x^{2}-33x+72 & =0 \\ x & =3 \text{ or } 8. \end{aligned}$$ Therefore, $AB=8$ and $AC=3$ or $AB=3$ and $AC=8$. In both cases, the area of the triangle is: $\frac{1}{2} \cdot 8 \cdot 3 \sin 60^{\circ}=6 \sqrt{3}$. | 0.875 | 3,278.375 | 2,576.428571 | 8,192 | [
{
"content": "Find the area of triangle $ABC$ given that $AB=8$, $AC=3$, and $\\angle BAC=60^{\\circ}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "6 \\sqrt{3}",
"style": "rule"
} | {
"index": 70,
"split": "easy"
} | [
{
"answer": "",
"count": 2,
"grade": 0
},
{
"answer": "6 \\sqrt{3}",
"count": 17,
"grade": 1
},
{
"answer": "6\\sqrt{3}",
"count": 13,
"grade": 1
}
] | |
0.6875 | 7,094.9375 | 6,596.272727 | 8,192 | [
{
"content": "Given a tetrahedron \\( P-ABC \\) with its four vertices on the surface of sphere \\( O \\), where \\( PA = PB = PC \\) and \\( \\triangle ABC \\) is an equilateral triangle with side length 2. \\( E \\) and \\( F \\) are the midpoints of \\( AC \\) and \\( BC \\) respectively, and \\( \\angle EPF... | math | {
"ground_truth": "6\\pi",
"style": "rule"
} | {
"index": 71,
"split": "easy"
} | [
{
"answer": "",
"count": 2,
"grade": 0
},
{
"answer": "6 \\pi",
"count": 2,
"grade": 1
},
{
"answer": "6\\pi",
"count": 56,
"grade": 1
}
] | ||
1. **Define Variables:**
Let $d$ be the distance Ian drove, $t$ be the time Ian drove, and $s$ be Ian's speed. Thus, Ian's equation is:
\[
d = s \cdot t
\]
This is equation (1).
2. **Set Up Han's Equation:**
Han drove 1 hour longer than Ian and at a speed 5 mph faster. Han also drove 70 miles more th... | 1 | 2,910.125 | 2,910.125 | -1 | [
{
"content": "Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian? Let's think step by step and outpu... | math | {
"ground_truth": "150",
"style": "rule"
} | {
"index": 72,
"split": "easy"
} | [
{
"answer": "",
"count": 1,
"grade": 0
},
{
"answer": "1.5 \\times 10^{2}",
"count": 1,
"grade": 0
},
{
"answer": "1.5 × 10²",
"count": 1,
"grade": 0
},
{
"answer": "150",
"count": 33,
"grade": 1
},
{
"answer": "150.0",
"count": 4,
"grade":... | |
1 | 2,065.5 | 2,065.5 | -1 | [
{
"content": "If $2^8=16^x$, find $x$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 73,
"split": "easy"
} | [
{
"answer": "2",
"count": 34,
"grade": 1
}
] | ||
0.75 | 5,051.5 | 4,004.666667 | 8,192 | [
{
"content": "The numbers from 1 to 200, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is neither a perfect square, a perfect cube, nor a sixth power? Express your answer as a common fraction. Let's think step by step and output the final answer within... | math | {
"ground_truth": "\\frac{183}{200}",
"style": "rule"
} | {
"index": 74,
"split": "easy"
} | [
{
"answer": "",
"count": 14,
"grade": 0
},
{
"answer": "\\frac{183}{200}",
"count": 37,
"grade": 1
},
{
"answer": "\\frac{61}{100}",
"count": 1,
"grade": 0
},
{
"answer": "\\frac{915}{1000}",
"count": 1,
"grade": 0
}
] | ||
0.875 | 5,894.1875 | 5,565.928571 | 8,192 | [
{
"content": "Jose invested $\\$50,\\!000$ for $2$ years at an annual interest rate of $4$ percent compounded yearly. Patricia invested $\\$50,\\!000$ for the same period of time, at the same interest rate, but the interest was compounded quarterly. To the nearest dollar, how much more money did Patricia's inve... | math | {
"ground_truth": "63",
"style": "rule"
} | {
"index": 75,
"split": "easy"
} | [
{
"answer": "63",
"count": 60,
"grade": 1
}
] | ||
0.875 | 5,079.4375 | 4,739.785714 | 7,457 | [
{
"content": "The positive four-digit integers that use each of the four digits $1,$ $2,$ $3,$ and $4$ exactly once are ordered from least to greatest. What is the $15^{\\text{th}}$ integer in the list? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "3214",
"style": "rule"
} | {
"index": 76,
"split": "easy"
} | [
{
"answer": "3214",
"count": 56,
"grade": 1
},
{
"answer": "\\frac{3214}{1}",
"count": 4,
"grade": 0
}
] | ||
0.9375 | 4,051.125 | 3,775.066667 | 8,192 | [
{
"content": "Convert \\(531_8\\) to base 7. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1002_7",
"style": "rule"
} | {
"index": 77,
"split": "easy"
} | [
{
"answer": "",
"count": 5,
"grade": 0
},
{
"answer": "1002 \\qc 7",
"count": 1,
"grade": 0
},
{
"answer": "1002 \\text{ base 7}",
"count": 1,
"grade": 0
},
{
"answer": "1002 \\text{ in base }7",
"count": 1,
"grade": 0
},
{
"answer": "1002\\_7",
... | ||
0.75 | 5,666.25 | 4,824.333333 | 8,192 | [
{
"content": "Given that $1 \\leq x \\leq 2$, calculate the number of different integer values of the expression $10-10|2x-3|$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "11",
"style": "rule"
} | {
"index": 78,
"split": "easy"
} | [
{
"answer": "",
"count": 5,
"grade": 0
},
{
"answer": "11",
"count": 48,
"grade": 1
},
{
"answer": "\\mi{11}",
"count": 1,
"grade": 1
},
{
"answer": "\\num{11}",
"count": 1,
"grade": 1
},
{
"answer": "\\overset{\\text{BMI}}{11}",
"count": 1,
... | ||
0.6875 | 425.875 | 423.909091 | 430.2 | [
{
"content": "Calculate $180 \\div \\left( 12 + 9 \\times 3 - 4 \\right)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{36}{7}",
"style": "rule"
} | {
"index": 79,
"split": "easy"
} | [
{
"answer": "",
"count": 16,
"grade": 0
},
{
"answer": "\\frac{36}{7}",
"count": 37,
"grade": 1
}
] | ||
0.9375 | 3,327.25 | 3,002.933333 | 8,192 | [
{
"content": "Consider a dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square, which is at a distance of 1 unit from the center, contains 8 unit squares. The second ring, at a distance of 2 units from the center, contains 16 unit squares. If the pa... | math | {
"ground_truth": "400",
"style": "rule"
} | {
"index": 80,
"split": "easy"
} | [
{
"answer": "4 \\, 10^2",
"count": 1,
"grade": 0
},
{
"answer": "4 \\times 100",
"count": 2,
"grade": 0
},
{
"answer": "4 \\times 10^2",
"count": 14,
"grade": 0
},
{
"answer": "4 \\times 10^{2}",
"count": 9,
"grade": 0
},
{
"answer": "4 \\times 10^... | ||
1. **Define the function $\heartsuit(x)$**: Given that $\heartsuit(x)$ is the average of $x$ and $x^2$, we use the formula for the average of two numbers $a$ and $b$, which is $\frac{a+b}{2}$. Therefore,
\[
\heartsuit(x) = \frac{x + x^2}{2}.
\]
2. **Calculate $\heartsuit(1)$**:
\[
\heartsuit(1) = \frac... | 1 | 2,443.375 | 2,443.375 | -1 | [
{
"content": "For a real number $x$, define $\\heartsuit(x)$ to be the average of $x$ and $x^2$. What is $\\heartsuit(1)+\\heartsuit(2)+\\heartsuit(3)$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 81,
"split": "easy"
} | [
{
"answer": "",
"count": 1,
"grade": 0
},
{
"answer": "10",
"count": 54,
"grade": 1
}
] | |
0.8125 | 5,190.75 | 5,067.153846 | 5,726.333333 | [
{
"content": "An isosceles triangle, a square, and a regular pentagon each have a perimeter of 20 inches. What is the ratio of the side length of the triangle (assuming both equal sides for simplicity) to the side length of the square? Express your answer as a common fraction. Let's think step by step and outpu... | math | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": 82,
"split": "easy"
} | [
{
"answer": "",
"count": 6,
"grade": 0
},
{
"answer": "4/3",
"count": 1,
"grade": 1
},
{
"answer": "4:3",
"count": 1,
"grade": 0
},
{
"answer": "\\dfrac{4}{3}",
"count": 1,
"grade": 1
},
{
"answer": "\\frac{4}{3}",
"count": 32,
"grade": 1
... | ||
1 | 2,826.375 | 2,826.375 | -1 | [
{
"content": "Simplify\n\\[(1 + \\cot A - \\csc A)(1 + \\tan A + \\sec A).\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 83,
"split": "easy"
} | [
{
"answer": "2",
"count": 42,
"grade": 1
}
] | ||
1. **Identify the given information:**
- The difference in the areas of two similar triangles is $18$ square feet.
- The ratio of the larger area to the smaller is the square of an integer.
- The area of the smaller triangle is an integer.
- One side of the smaller triangle is $3$ feet.
2. **Set up the equ... | 1 | 1,968.5 | 1,968.5 | -1 | [
{
"content": "The difference in the areas of two similar triangles is $18$ square feet, and the ratio of the larger area to the smaller is the square of an integer. The area of the smaller triangle, in square feet, is an integer, and one of its sides is $3$ feet. The corresponding side of the larger triangle,... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 84,
"split": "easy"
} | [
{
"answer": "",
"count": 5,
"grade": 0
},
{
"answer": "6",
"count": 16,
"grade": 1
},
{
"answer": "6.0",
"count": 1,
"grade": 1
},
{
"answer": "6OfWork",
"count": 1,
"grade": 0
},
{
"answer": "\\$6\\$",
"count": 1,
"grade": 1
}
] | |
1 | 2,808.8125 | 2,808.8125 | -1 | [
{
"content": "It is known that the numbers $\\frac{x}{2}, 2x - 3, \\frac{18}{x} + 1$, taken in the specified order, form a geometric progression. Find the common ratio of this progression. Round your answer to two decimal places. Let's think step by step and output the final answer within \\boxed{}.",
"role... | math | {
"ground_truth": "2.08",
"style": "rule"
} | {
"index": 85,
"split": "easy"
} | [
{
"answer": "2 \\, \\text{and} \\, 8 \\, \\text{tenths}",
"count": 1,
"grade": 0
},
{
"answer": "2.08",
"count": 24,
"grade": 1
},
{
"answer": "2.08\\,\\text{unit}",
"count": 1,
"grade": 0
},
{
"answer": "2.08\\footnotesize\\Order\\ of\\ magnitude",
"count": 1... | ||
0.875 | 4,199.625 | 3,629.285714 | 8,192 | [
{
"content": "What is the smallest positive integer $n$ such that $2n$ is a perfect square and $3n$ is a perfect cube? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "72",
"style": "rule"
} | {
"index": 86,
"split": "easy"
} | [
{
"answer": "72",
"count": 35,
"grade": 1
},
{
"answer": "\\ 72",
"count": 2,
"grade": 1
}
] | ||
$\dfrac{\cos{96^{\circ}}+\sin{96^{\circ}}}{\cos{96^{\circ}}-\sin{96^{\circ}}} = \dfrac{1 + \tan{96^{\circ}}}{1-\tan{96^{\circ}}}$ which is the same as $\dfrac{\tan{45^{\circ}} + \tan{96^{\circ}}}{1-\tan{45^{\circ}}\tan{96^{\circ}}} = \tan{141{^\circ}}$.
So $19x = 141 +180n$, for some integer $n$. Multiplying by $19$ g... | 0.875 | 5,773.6875 | 5,428.214286 | 8,192 | [
{
"content": "Find the smallest positive integer solution to $\\tan{19x^{\\circ}}=\\dfrac{\\cos{96^{\\circ}}+\\sin{96^{\\circ}}}{\\cos{96^{\\circ}}-\\sin{96^{\\circ}}}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "159",
"style": "rule"
} | {
"index": 87,
"split": "easy"
} | [
{
"answer": "",
"count": 5,
"grade": 0
},
{
"answer": "159",
"count": 54,
"grade": 1
},
{
"answer": "159 многократно",
"count": 1,
"grade": 0
}
] | |
0.9375 | 5,000.9375 | 4,788.2 | 8,192 | [
{
"content": "Given the line $l: 4x+3y-8=0$ passes through the center of the circle $C: x^2+y^2-ax=0$ and intersects circle $C$ at points A and B, with O as the origin.\n(I) Find the equation of circle $C$.\n(II) Find the equation of the tangent to circle $C$ at point $P(1, \\sqrt {3})$.\n(III) Find the area of... | math | {
"ground_truth": "\\frac{16}{5}",
"style": "rule"
} | {
"index": 88,
"split": "easy"
} | [
{
"answer": "",
"count": 7,
"grade": 0
},
{
"answer": "16/5",
"count": 1,
"grade": 1
},
{
"answer": "\\frac{16}{5}",
"count": 44,
"grade": 1
},
{
"answer": "\\frac{16}{5}\"",
"count": 1,
"grade": 0
},
{
"answer": "\\small\\frac{16}{5}",
"count"... | ||
0.9375 | 2,854.3125 | 2,778.466667 | 3,992 | [
{
"content": "Let \\\\(f(x)=ax^{2}-b\\sin x\\\\) and \\\\(f′(0)=1\\\\), \\\\(f′\\left( \\dfrac {π}{3}\\right)= \\dfrac {1}{2}\\\\). Find the values of \\\\(a\\\\) and \\\\(b\\\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": 89,
"split": "easy"
} | [
{
"answer": "-1",
"count": 32,
"grade": 1
},
{
"answer": "-1_EDGE_QUERY",
"count": 1,
"grade": 0
}
] | ||
0.8125 | 4,388.9375 | 3,511.307692 | 8,192 | [
{
"content": "What is the value of $\\frac{14!}{5!9!}$ ? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2002",
"style": "rule"
} | {
"index": 90,
"split": "easy"
} | [
{
"answer": "2002",
"count": 51,
"grade": 1
},
{
"answer": "2002สาธารอก но clockwise",
"count": 1,
"grade": 0
}
] | ||
1 | 3,958.9375 | 3,958.9375 | -1 | [
{
"content": "$A, B, C, D,$ and $E$ are collinear in that order such that $AB = BC = 1, CD = 2,$ and $DE = 9$. If $P$ can be any point in space, what is the smallest possible value of $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$?\n Let's think step by step and output the final answer within \\boxed{}.",
"role": "user... | math | {
"ground_truth": "110",
"style": "rule"
} | {
"index": 91,
"split": "easy"
} | [
{
"answer": "110",
"count": 49,
"grade": 1
},
{
"answer": "110 naïve",
"count": 1,
"grade": 0
}
] | ||
0.8125 | 3,457 | 2,741.846154 | 6,556 | [
{
"content": "Lucy has $19$ dollars and $23$ cents. She wants to buy as many popsicles as she can with her money. The popsicles are priced at $1$ dollar and $60$ cents each. How many popsicles can she buy? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 92,
"split": "easy"
} | [
{
"answer": "",
"count": 3,
"grade": 0
},
{
"answer": "12",
"count": 56,
"grade": 1
},
{
"answer": "12六",
"count": 1,
"grade": 0
}
] | ||
Observe that $$\frac{1}{x+yi}=\frac{x-yi}{x^{2}+y^{2}}=33x-56y+(56x+33y)i=(33+56i)(x+yi)$$ So $$(x+yi)^{2}=\frac{1}{33+56i}=\frac{1}{(7+4i)^{2}}=\left(\frac{7-4i}{65}\right)^{2}$$ It follows that $(x, y)= \pm\left(\frac{7}{65},-\frac{4}{65}\right)$. | 1 | 4,721.25 | 4,721.25 | -1 | [
{
"content": "Let $(x, y)$ be a pair of real numbers satisfying $$56x+33y=\\frac{-y}{x^{2}+y^{2}}, \\quad \\text { and } \\quad 33x-56y=\\frac{x}{x^{2}+y^{2}}$$ Determine the value of $|x|+|y|$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{11}{65}",
"style": "rule"
} | {
"index": 93,
"split": "easy"
} | [
{
"answer": "",
"count": 7,
"grade": 0
},
{
"answer": "\\frac{11}{65}",
"count": 43,
"grade": 1
}
] | |
0.75 | 4,037.625 | 3,840.5 | 4,629 | [
{
"content": "For the Olympic torch relay, it is planned to select 6 cities from 8 in a certain province to establish the relay route, satisfying the following conditions. How many methods are there for each condition?\n(1) Only one of the two cities, A and B, is selected. How many methods are there? How many d... | math | {
"ground_truth": "19440",
"style": "rule"
} | {
"index": 94,
"split": "easy"
} | [
{
"answer": "19440",
"count": 60,
"grade": 1
}
] | ||
0.6875 | 2,541.625 | 2,749.636364 | 2,084 | [
{
"content": "Evaluate $\\lfloor-5.77\\rfloor+\\lceil-3.26\\rceil+\\lfloor15.93\\rfloor+\\lceil32.10\\rceil$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "39",
"style": "rule"
} | {
"index": 95,
"split": "easy"
} | [
{
"answer": "",
"count": 1,
"grade": 0
},
{
"answer": "39",
"count": 42,
"grade": 1
},
{
"answer": "\\displaystyle 39",
"count": 1,
"grade": 1
}
] | ||
0.75 | 5,671.5 | 5,172.25 | 7,169.25 | [
{
"content": "Calculate:<br/>$(1)(\\sqrt{\\frac{1}{3}})^{2}+\\sqrt{0．{3}^{2}}-\\sqrt{\\frac{1}{9}}$;<br/>$(2)(\\sqrt{6}-\\sqrt{\\frac{1}{2}})-(\\sqrt{24}+2\\sqrt{\\frac{2}{3}})$;<br/>$(3)(\\frac{\\sqrt{32}}{3}-4\\sqrt{\\frac{1}{2}}+3\\sqrt{27})÷2\\sqrt{2}$;<br/>$(4)(\\sqrt{3}+\\sqrt{2}-1)(\\sqrt{3}-\\sqrt{2}+1)... | math | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": 96,
"split": "easy"
} | [
{
"answer": "",
"count": 1,
"grade": 0
},
{
"answer": "2 \\sqrt{2}",
"count": 13,
"grade": 1
},
{
"answer": "2\\sqrt{2}",
"count": 39,
"grade": 1
}
] | ||
1 | 1,999.3125 | 1,999.3125 | -1 | [
{
"content": "Let $\\mathbf{P}$ be the matrix for projecting onto the vector $\\begin{pmatrix} 4 \\\\ -7 \\end{pmatrix}.$ Find $\\det \\mathbf{P}.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 97,
"split": "easy"
} | [
{
"answer": "",
"count": 2,
"grade": 0
},
{
"answer": "0",
"count": 18,
"grade": 1
},
{
"answer": "var 0;",
"count": 1,
"grade": 0
}
] | ||
0.6875 | 6,543.6875 | 5,794.454545 | 8,192 | [
{
"content": "Pentagon ABCDE has a vertical line of symmetry. What is the $y$-coordinate of vertex C so that the area of the pentagon is 40 square units? [asy]\nunitsize(2mm);\ndefaultpen(linewidth(.7pt)+fontsize(8pt));\ndotfactor=4;\n\npair A=(0,0), B=(0,4), C=(2,16), D=(4,4), E=(4,0);\npair[] dots={A,B,C,D,E}... | math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": 98,
"split": "easy"
} | [
{
"answer": "",
"count": 5,
"grade": 0
},
{
"answer": "$16$独自 regardless of implementation",
"count": 1,
"grade": 0
},
{
"answer": "16",
"count": 24,
"grade": 1
},
{
"answer": "16 y o r l …",
"count": 1,
"grade": 0
},
{
"answer": "2^4",
"count... | ||
1 | 1,889.9375 | 1,889.9375 | -1 | [
{
"content": "If $(x+2)(x-3)=14$, find the product of the possible values of $x$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-20",
"style": "rule"
} | {
"index": 99,
"split": "easy"
} | [
{
"answer": "-20",
"count": 60,
"grade": 1
}
] |
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