data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In the expansion of $(x+y)(x-y)^{5}$, the coefficient of $x^{2}y^{4}$ is ____ (provide your answer as a number). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "39900",
"split": "train"
} |
math | [
{
"content": "Let $P$ be a point on the line\n\\[\\begin{pmatrix} 3 \\\\ -1 \\\\ 2 \\end{pmatrix} + t \\begin{pmatrix} 2 \\\\ -2 \\\\ 1 \\end{pmatrix}\\]and let $Q$ be a point on the line\n\\[\\begin{pmatrix} 0 \\\\ 0 \\\\ 4 \\end{pmatrix} + s \\begin{pmatrix} 1 \\\\ 2 \\\\ -1 \\end{pmatrix}.\\]Find the shortes... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "39901",
"split": "train"
} |
math | [
{
"content": "If $8^{2x} = 11$, evaluate $2^{x + 1.5}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11^{1/6} \\cdot 2 \\sqrt{2}",
"style": "rule"
} | {
"index": "39902",
"split": "train"
} |
math | [
{
"content": "$ABCDE$ is a regular pentagon inscribed in a circle of radius 1. What is the area of the set of points inside the circle that are farther from $A$ than they are from any other vertex? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{5}",
"style": "rule"
} | {
"index": "39903",
"split": "train"
} |
math | [
{
"content": "A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of $4$, and the group must have at least one singer. Let $N$ be the number of different groups that cou... | {
"ground_truth": "95",
"style": "rule"
} | {
"index": "39904",
"split": "train"
} |
math | [
{
"content": "Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences. Please output the final answer within ... | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "39905",
"split": "train"
} |
math | [
{
"content": "Quadrilateral $ABCD$ has right angles at $B$ and $D$. The length of the diagonal $AC$ is $5$. If two sides of $ABCD$ have integer lengths and one of these lengths is an odd integer, determine the area of $ABCD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39906",
"split": "train"
} |
math | [
{
"content": "Calculate using your preferred method: \n(1) $42.67-(12.95-7.33)$ \n(2) $\\left[8.4-8.4\\times(3.12-3.7)\\right]\\div0.42$ \n(3) $5.13\\times0.23+8.7\\times0.513-5.13$ \n(4) $6.66\\times222+3.33\\times556$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3330",
"style": "rule"
} | {
"index": "39907",
"split": "train"
} |
math | [
{
"content": "What is the greatest integer value of $b$ such that $-4$ is not in the range of $y=x^2+bx+12$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "39908",
"split": "train"
} |
math | [
{
"content": "The letters $A$, $B$, $C$ and $D$ represent digits. If $\\begin{array}{ccc}&A&B\\\\ +&C&A\\\\ \\hline &D&A\\end{array}$and $\\begin{array}{ccc}&A&B\\\\ -&C&A\\\\ \\hline &&A\\end{array}$,what digit does $D$ represent? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "39909",
"split": "train"
} |
math | [
{
"content": "The distance between two parallel lines $x-y=1$ and $2x-2y+3=0$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "39910",
"split": "train"
} |
math | [
{
"content": "In order to cultivate students' financial management skills, Class 1 of the second grade founded a \"mini bank\". Wang Hua planned to withdraw all the money from a deposit slip. In a hurry, the \"bank teller\" mistakenly swapped the integer part (the amount in yuan) with the decimal part (the amou... | {
"ground_truth": "14.32",
"style": "rule"
} | {
"index": "39911",
"split": "train"
} |
math | [
{
"content": "Find $\\cos C$ in the right triangle shown below.\n\n\n[asy]\npair A,B,C;\nA = (0,0);\nB = (6,0);\nC = (0,8);\ndraw(A--B--C--A);\ndraw(rightanglemark(B,A,C,10));\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,SE);\nlabel(\"$C$\",C,N);\nlabel(\"$10$\",(B+C)/2,NE);\nlabel(\"$6$\",B/2,S);\n[/asy] Please outp... | {
"ground_truth": "\\frac45",
"style": "rule"
} | {
"index": "39912",
"split": "train"
} |
math | [
{
"content": "Circles of radius 3 and 4 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24\\pi",
"style": "rule"
} | {
"index": "39913",
"split": "train"
} |
math | [
{
"content": "I bought a lottery ticket with a five-digit number such that the sum of its digits equals the age of my neighbor. Determine the number of this ticket, given that my neighbor easily solved this problem. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "99999",
"style": "rule"
} | {
"index": "39914",
"split": "train"
} |
math | [
{
"content": "Two right triangles, $ABC$ and $ACD$, are joined as shown. Squares are drawn on four of the sides. The areas of three of the squares are 9, 16 and 36 square units. What is the number of square units in the area of the fourth square?\n\nNote that the diagram is not drawn to scale.\n\n[asy]\ndefault... | {
"ground_truth": "61",
"style": "rule"
} | {
"index": "39915",
"split": "train"
} |
math | [
{
"content": "Nine lines parallel to the base of a triangle divide the other sides each into $10$ equal segments and the area into $10$ distinct parts. If the area of the largest of these parts is $38$ , then the area of the original triangle is Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "39916",
"split": "train"
} |
math | [
{
"content": "The density of a body is defined as the ratio of its mass to the volume it occupies. There is a homogeneous cube with a volume of $V=8 \\, m^{3}$. As a result of heating, each of its edges increased by 4 mm. By what percentage did the density of this cube change? Please output the final answer wit... | {
"ground_truth": "0.6",
"style": "rule"
} | {
"index": "39917",
"split": "train"
} |
math | [
{
"content": "Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\\circ$ about its center and the top sheet is rotated clockwise $60^\\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. ... | {
"ground_truth": "147",
"style": "rule"
} | {
"index": "39918",
"split": "train"
} |
math | [
{
"content": "Cyclic quadrilateral $A B C D$ has side lengths $A B=1, B C=2, C D=3$ and $D A=4$. Points $P$ and $Q$ are the midpoints of $\\overline{B C}$ and $\\overline{D A}$. Compute $P Q^{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{116}{35}",
"style": "rule"
} | {
"index": "39919",
"split": "train"
} |
math | [
{
"content": "Find all real numbers $x$ such that\n\\[\\frac{8^x + 27^x}{12^x + 18^x} = \\frac{7}{6}.\\]Enter all the solutions, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1,1",
"style": "rule"
} | {
"index": "39920",
"split": "train"
} |
math | [
{
"content": "The equation\n$x^{10}+(13x-1)^{10}=0\\,$\n has 10 complex roots $r_1, \\overline{r_1}, r_2, \\overline{r_2}, r_3, \\overline{r_3}, r_4, \\overline{r_4}, r_5, \\overline{r_5},\\,$ where the bar denotes complex conjugation. Find the value of\n$\\frac 1{r_1\\overline{r_1}}+\\frac 1{r_2\\overline{r_2}... | {
"ground_truth": "850",
"style": "rule"
} | {
"index": "39921",
"split": "train"
} |
math | [
{
"content": "Given a triangle with sides of lengths 3, 4, and 5. Three circles are constructed with radii of 1, centered at each vertex of the triangle. Find the total area of the portions of the circles that lie within the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "39922",
"split": "train"
} |
math | [
{
"content": "Given $\\tan\\left( \\frac{\\pi}{4} + \\alpha \\right) = \\frac{1}{7}$, with $\\alpha \\in \\left( \\frac{\\pi}{2}, \\pi \\right)$, find the value of $\\tan\\alpha$ and $\\cos\\alpha$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{4}{5}",
"style": "rule"
} | {
"index": "39923",
"split": "train"
} |
math | [
{
"content": "During the Qingming Festival, a certain school, in order to commemorate the revolutionary martyrs, requires students to participate in the \"Qingming Sacrifice to the Martyrs\" activity by either visiting the Revolutionary Martyrs Memorial Hall or participating online. Students can only choose one... | {
"ground_truth": "\\frac{5}{21}",
"style": "rule"
} | {
"index": "39924",
"split": "train"
} |
math | [
{
"content": "Find the product of the roots of the equation $24x^2 + 60x - 750 = 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{125}{4}",
"style": "rule"
} | {
"index": "39925",
"split": "train"
} |
math | [
{
"content": "Five identical balls roll on a smooth horizontal surface towards each other. The velocities of the first and second are $v_{1}=v_{2}=0.5$ m/s, and the velocities of the others are $v_{3}=v_{4}=v_{5}=0.1$ m/s. The initial distances between the balls are the same, $l=2$ m. All collisions are perfect... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39926",
"split": "train"
} |
math | [
{
"content": "We are given triangle $A B C$, with $A B=9, A C=10$, and $B C=12$, and a point $D$ on $B C . B$ and $C$ are reflected in $A D$ to $B^{\\prime}$ and $C^{\\prime}$, respectively. Suppose that lines $B C^{\\prime}$ and $B^{\\prime} C$ never meet (i.e., are parallel and distinct). Find $B D$. Please o... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "39927",
"split": "train"
} |
math | [
{
"content": "Twenty-five cards are randomly arranged in a grid. Five of these cards have a 0 on one side and a 1 on the other side. The remaining twenty cards either have a 0 on both sides or a 1 on both sides. Loron chooses one row or one column and flips over each of the five cards in that row or column, lea... | {
"ground_truth": "9:16",
"style": "rule"
} | {
"index": "39928",
"split": "train"
} |
math | [
{
"content": "Square A has side lengths each measuring $x$ inches. Square B has side lengths each measuring $5x$ inches. Square C has side lengths each measuring $2x$ inches. What is the ratio of the area of Square A to the area of Square B, and what is the ratio of the area of Square C to the area of Square B?... | {
"ground_truth": "\\frac{4}{25}",
"style": "rule"
} | {
"index": "39929",
"split": "train"
} |
math | [
{
"content": "A large supermarket has recorded the number of customers during eight holidays (unit: hundreds of people) as $29$, $30$, $38$, $25$, $37$, $40$, $42$, $32$. What is the $75$th percentile of this data set? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "39930",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has sides of length 5, 12 and 13 units, and triangle $DEF$ has sides of length 8, 15 and 17 units. What is the ratio of the area of triangle $ABC$ to the area of triangle $DEF$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "\\dfrac{1}{2}",
"style": "rule"
} | {
"index": "39931",
"split": "train"
} |
math | [
{
"content": "Given complex numbers ${z_1}=2i$, ${z_2}=1-i$, where $i$ is the imaginary unit,\n\n(1) Find the conjugate of the complex number $\\frac{z_1}{z_2}$;\n\n(2) In the complex plane, let points $Z_1$, $Z_2$ correspond to ${z_1}$, ${z_2}$ respectively, and $O$ be the origin. Form a parallelogram with $\\... | {
"ground_truth": "\\sqrt{10}",
"style": "rule"
} | {
"index": "39932",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system, define $d(P, Q) = |x_1 - x_2| + |y_1 - y_2|$ as the \"polyline distance\" between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$. Then, the minimum \"polyline distance\" between a point on the circle $x^2 + y^2 = 1$ and a point on the line $2x + y - 2 \\sqrt{5} = 0$... | {
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "39933",
"split": "train"
} |
math | [
{
"content": "Using systematic sampling to select a sample of size 20 from 180 students, the students are randomly numbered from 1 to 180. They are then divided into 20 groups in order of their number (group 1: numbers 1-9, group 2: numbers 10-18, ..., group 20: numbers 172-180). If the number drawn from group ... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "39934",
"split": "train"
} |
math | [
{
"content": "We flip a fair coin 12 times. What is the probability that we get exactly 9 heads and all heads occur consecutively? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{1024}",
"style": "rule"
} | {
"index": "39935",
"split": "train"
} |
math | [
{
"content": "In the sequence\n\\[..., a, b, c, d, 0, 1, 1, 2, 3, 5, 8,...\\]\neach term is the sum of the two terms to its left. Find $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "39936",
"split": "train"
} |
math | [
{
"content": "From the numbers \\\\(1\\\\), \\\\(2\\\\), \\\\(3\\\\), \\\\(4\\\\), \\\\(5\\\\), \\\\(6\\\\), two numbers are selected to form an ordered pair of real numbers \\\\((x, y)\\\\). The probability that \\\\(\\dfrac{x}{y+1}\\\\) is an integer is equal to \\_\\_\\_\\_\\_\\_ Please output the final answ... | {
"ground_truth": "\\dfrac{4}{15}",
"style": "rule"
} | {
"index": "39937",
"split": "train"
} |
math | [
{
"content": "A function $g$ is defined by $g(z) = (3 + 2i) z^2 + \\beta z + \\delta$ for all complex numbers $z$, where $\\beta$ and $\\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\\beta| + |\\delta|$? Please output the fina... | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "39938",
"split": "train"
} |
math | [
{
"content": "In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \\perp BC, BE \\perp CA, CF \\perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \\perp EF$ and $HQ \\perp EF$. Lines $DP$ and $QH$ ... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39939",
"split": "train"
} |
math | [
{
"content": "A spherical decoration was suspended in a cylindrical container when the water inside it froze. The decoration was removed (without breaking the ice), leaving a hole 30 cm across at the top and 10 cm deep. If the water froze up to a height of 5 cm from the top of the sphere, what was the radius of... | {
"ground_truth": "5\\sqrt{13}",
"style": "rule"
} | {
"index": "39940",
"split": "train"
} |
math | [
{
"content": "Let $(p_n)$ and $(q_n)$ be sequences of real numbers defined by the equation\n\\[ (3+i)^n = p_n + q_n i\\] for all integers $n \\geq 0$, where $i = \\sqrt{-1}$. Determine the value of\n\\[ \\sum_{n=0}^\\infty \\frac{p_n q_n}{9^n}.\\]\nA) $\\frac{5}{8}$\nB) $\\frac{7}{8}$\nC) $\\frac{3}{4}$\nD) $\\... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "39941",
"split": "train"
} |
math | [
{
"content": "Evaluate $(x-a+3)$ if the value of $x$ is $(a+7)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39942",
"split": "train"
} |
math | [
{
"content": "The hypotenuse of a right triangle measures $6\\sqrt{2}$ inches and one angle is $45^{\\circ}$. What is the number of square inches in the area of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "39943",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $BC = 4$, $AC = 3 \\sqrt{2}$, and $\\angle C = 45^\\circ$. Altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$. Find $AH:HD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39944",
"split": "train"
} |
math | [
{
"content": "How many 3-digit numbers have the property that the units digit is at least twice the tens digit? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "270",
"style": "rule"
} | {
"index": "39945",
"split": "train"
} |
math | [
{
"content": "Given that $b$ is a multiple of $570$, find the greatest common divisor of $4b^3 + 2b^2 + 5b + 171$ and $b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "171",
"style": "rule"
} | {
"index": "39946",
"split": "train"
} |
math | [
{
"content": "Given real numbers $x$ and $y$ satisfy the equation $x^2+y^2-4x+1=0$.\n(1) Find the maximum and minimum value of $\\frac {y}{x}$.\n(2) Find the maximum and minimum value of $y-x$.\n(3) Find the maximum and minimum value of $x^2+y^2$. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "7-4\\sqrt{3}",
"style": "rule"
} | {
"index": "39947",
"split": "train"
} |
math | [
{
"content": "James wrote a different integer from 1 to 9 in each cell of a table. He then calculated the sum of the integers in each of the rows and in each of the columns of the table. Five of his answers were 12, 13, 15, 16, and 17, in some order. What was his sixth answer? Please output the final answer wit... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "39948",
"split": "train"
} |
math | [
{
"content": "Select 4 shoes from 5 pairs of different-sized shoes, find the number of possibilities where at least 2 of the 4 shoes can be paired together. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "130",
"style": "rule"
} | {
"index": "39949",
"split": "train"
} |
math | [
{
"content": "A natural number \\( n \\) is called \"cubowat\" if \\( n^{3} + 13n - 273 \\) is a cube of a natural number. Find the sum of all cubowat numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "39950",
"split": "train"
} |
math | [
{
"content": "The ratio of the number of games won to the number of games lost by the High School Hurricanes is $7/3$ with 5 games ended in a tie. Determine the percentage of games lost by the Hurricanes, rounded to the nearest whole percent. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "24\\%",
"style": "rule"
} | {
"index": "39951",
"split": "train"
} |
math | [
{
"content": "How many eight-digit numbers can be written using only the digits 1, 2, and 3 such that the difference between any two adjacent digits is 1? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "39952",
"split": "train"
} |
math | [
{
"content": "Sara used $\\frac{3}{10}$ of a roll of wrapping paper to wrap three presents. If she used the same amount of wrapping paper on each present, what fraction of a roll of wrapping paper did she use on each present? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{10}",
"style": "rule"
} | {
"index": "39953",
"split": "train"
} |
math | [
{
"content": "There are 3 male students and 2 female students. Calculate the total number of different arrangement methods under the following different requirements: \n(1) All are arranged in a row where \"A\" can only be in the middle or on the sides. There are \\_\\_\\_\\_\\_\\_ ways to arrange this; \n(2... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "39954",
"split": "train"
} |
math | [
{
"content": "The angle $A$ at the vertex of the isosceles triangle $ABC$ is $100^{\\circ}$. On the ray $AB$, a segment $AM$ is laid off, equal to the base $BC$. Find the measure of the angle $BCM$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39955",
"split": "train"
} |
math | [
{
"content": "In square ABCD, where AB=2, fold along the diagonal AC so that plane ABC is perpendicular to plane ACD, resulting in the pyramid B-ACD. Find the ratio of the volume of the circumscribed sphere of pyramid B-ACD to the volume of pyramid B-ACD. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "4\\pi:1",
"style": "rule"
} | {
"index": "39956",
"split": "train"
} |
math | [
{
"content": "Determine the maximum integer $ n $ such that for each positive integer $ k \\le \\frac{n}{2} $ there are two positive divisors of $ n $ with difference $ k $. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "39957",
"split": "train"
} |
math | [
{
"content": "What is the product of the prime numbers less than 20? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9699690",
"style": "rule"
} | {
"index": "39958",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)= \\begin{cases} \\left( \\frac {1}{2}\\right)^{x} & (x\\geqslant 4) \\\\ f(x+1) & (x < 4)\\end{cases}$, find the value of $f(\\log _{2}3)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{24}",
"style": "rule"
} | {
"index": "39959",
"split": "train"
} |
math | [
{
"content": "A 5x5x5 cube is formed by assembling 125 unit cubes. Nine unit squares are painted on each of the six faces of the cube according to the pattern shown. How many of the 125 unit cubes have no paint on them? [asy]\nfill((2,0)--(3,0)--(3,5)--(2,5)--cycle,gray(.7));\nfill((0,2)--(0,3)--(5,3)--(5,2)--c... | {
"ground_truth": "83",
"style": "rule"
} | {
"index": "39960",
"split": "train"
} |
math | [
{
"content": "As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour, and Emerson skates ... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "39961",
"split": "train"
} |
math | [
{
"content": "The equation of the line shown can be written as $y=mx+b$. Find $mb$.\n[asy]\nsize(100,0);\nadd(shift(-5,-5)*grid(10,10));\ndraw((-5,0)--(5,0),linewidth(2));\ndraw((0,-5)--(0,5),linewidth(2));\nlabel(\"\",(5,0),E);\nlabel(\"\",(0,5),N);\ndraw((-3,-5) -- (2,5),blue,Arrows);\n[/asy]\nEach grid squa... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39962",
"split": "train"
} |
math | [
{
"content": "The cafe \"Burattino\" operates 6 days a week with a day off on Mondays. Kolya made two statements: \"from April 1 to April 20, the cafe worked 18 days\" and \"from April 10 to April 30, the cafe also worked 18 days.\" It is known that he made a mistake once. How many days did the cafe work from A... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "39963",
"split": "train"
} |
math | [
{
"content": "Xiao Li drove from location A to location B. Two hours after departure, the car broke down at location C, and it took 40 minutes to repair. After the repair, the speed was only 75% of the normal speed, resulting in arrival at location B being 2 hours later than planned. If the car had instead brok... | {
"ground_truth": "288",
"style": "rule"
} | {
"index": "39964",
"split": "train"
} |
math | [
{
"content": "For how many values of $k$ is $60^{10}$ the least common multiple of the positive integers $10^{10}$, $12^{12}$, and $k$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "121",
"style": "rule"
} | {
"index": "39965",
"split": "train"
} |
math | [
{
"content": "The repeating decimal for $\\frac{5}{13}$ is $0.cdc\\ldots$ What is the value of the sum $c+d$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "39966",
"split": "train"
} |
math | [
{
"content": "Inside a unit cube, eight equal spheres are placed. Each sphere is inscribed in one of the trihedral angles of the cube and touches the three spheres corresponding to the adjacent vertices of the cube. Find the radii of the spheres. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "1/4",
"style": "rule"
} | {
"index": "39967",
"split": "train"
} |
math | [
{
"content": "Juan takes a number, adds 3 to it, squares the result, then multiplies the answer by 2, subtracts 3 from the result, and finally divides that number by 2. If his final answer is 49, what was the original number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{\\frac{101}{2}} - 3",
"style": "rule"
} | {
"index": "39968",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{v} = \\begin{pmatrix} 4 \\\\ -5 \\end{pmatrix}$ and $\\mathbf{w} = \\begin{pmatrix} 13 \\\\ 2 \\end{pmatrix}$. Find the area of the parallelogram formed by vectors $\\mathbf{v}$ and $2\\mathbf{w}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "146",
"style": "rule"
} | {
"index": "39969",
"split": "train"
} |
math | [
{
"content": "Find the number of different possible rational roots of the polynomial:\n\\[6x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 10 = 0.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "39970",
"split": "train"
} |
math | [
{
"content": "If $2\\tan\\alpha=3\\tan \\frac{\\pi}{8}$, then $\\tan\\left(\\alpha- \\frac{\\pi}{8}\\right)=$ ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\sqrt{2}+1}{49}",
"style": "rule"
} | {
"index": "39971",
"split": "train"
} |
math | [
{
"content": "What's the largest number of elements that a set of positive integers between $1$ and $100$ inclusive can have if it has the property that none of them is divisible by another? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "39972",
"split": "train"
} |
math | [
{
"content": "Find the largest possible value of $x$ in the simplified form $x=\\frac{a+b\\sqrt{c}}{d}$ if $\\frac{5x}{6}+1=\\frac{3}{x}$, where $a,b,c,$ and $d$ are integers. What is $\\frac{acd}{b}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-55",
"style": "rule"
} | {
"index": "39973",
"split": "train"
} |
math | [
{
"content": "Alice thinks of four positive integers $a \\leq b \\leq c \\leq d$ satisfying $\\{a b+c d, a c+b d, a d+b c\\}=\\{40,70,100\\}$. What are all the possible tuples $(a, b, c, d)$ that Alice could be thinking of? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(1,4,6,16)",
"style": "rule"
} | {
"index": "39974",
"split": "train"
} |
math | [
{
"content": "\\(ABCD\\) is a square-based pyramid with base \\(ABCD\\) and apex \\(E\\). Point \\(E\\) is directly above point \\(A\\), with \\(AE = 1024\\) units and \\(AB = 640\\) units. The pyramid is sliced into two parts by a horizontal plane parallel to the base \\(ABCD\\), at a height \\(h\\) above the ... | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "39975",
"split": "train"
} |
math | [
{
"content": "In the diagram, $JKLM$ and $NOPM$ are squares each of area 25. If $Q$ is the midpoint of both $KL$ and $NO$, find the total area of polygon $JMQPON$.\n\n[asy]\nunitsize(3 cm);\n\npair J, K, L, M, N, O, P, Q;\n\nO = (0,0);\nP = (1,0);\nM = (1,1);\nN = (0,1);\nQ = (N + O)/2;\nJ = reflect(M,Q)*(P);\n... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "39976",
"split": "train"
} |
math | [
{
"content": "In a round-robin chess tournament, 30 players are participating. To achieve the 4th category norm, a player needs to score 60% of the possible points. What is the maximum number of players who can achieve the category norm by the end of the tournament? Please output the final answer within \\boxed... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "39977",
"split": "train"
} |
math | [
{
"content": "For each positive integer $n$, define $S(n)$ to be the smallest positive integer divisible by each of the positive integers $1, 2, 3, \\ldots, n$. How many positive integers $n$ with $1 \\leq n \\leq 100$ have $S(n) = S(n+4)$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "39978",
"split": "train"
} |
math | [
{
"content": "Rectangle ABCD and right triangle AEF share side AD and have the same area. Side AD = 8, and side AB = 7. If EF, which is perpendicular to AD, is denoted as x, determine the length of hypotenuse AF. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{65}",
"style": "rule"
} | {
"index": "39979",
"split": "train"
} |
math | [
{
"content": "Team A and Team B have a table tennis team match. Each team has three players, and each player plays once. Team A's three players are \\( A_{1}, A_{2}, A_{3} \\) and Team B's three players are \\( B_{1}, B_{2}, B_{3} \\). The winning probability of \\( A_{i} \\) against \\( B_{j} \\) is \\( \\frac... | {
"ground_truth": "91/60",
"style": "rule"
} | {
"index": "39980",
"split": "train"
} |
math | [
{
"content": "Triangle $A B C$ has side lengths $A B=15, B C=18, C A=20$. Extend $C A$ and $C B$ to points $D$ and $E$ respectively such that $D A=A B=B E$. Line $A B$ intersects the circumcircle of $C D E$ at $P$ and $Q$. Find the length of $P Q$. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "39981",
"split": "train"
} |
math | [
{
"content": "How many integers $-12 \\leq n \\leq 12$ satisfy $(n-3)(n+5)(n+9)<0$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39982",
"split": "train"
} |
math | [
{
"content": "50 businessmen - Japanese, Koreans, and Chinese - are sitting at a round table. It is known that between any two nearest Japanese, there are exactly as many Chinese as there are Koreans at the table. How many Chinese can be at the table? Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "39983",
"split": "train"
} |
math | [
{
"content": "The inverse of \\( f(x) = \\frac{3x-2}{x+4} \\) may be written in the form \\( f^{-1}(x) = \\frac{ax+b}{cx+d} \\), where \\( a, b, c, \\) and \\( d \\) are real numbers. Find \\( \\frac{a}{c} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "39984",
"split": "train"
} |
math | [
{
"content": "In triangle $A B C, \\angle B A C=60^{\\circ}$. Let \\omega be a circle tangent to segment $A B$ at point $D$ and segment $A C$ at point $E$. Suppose \\omega intersects segment $B C$ at points $F$ and $G$ such that $F$ lies in between $B$ and $G$. Given that $A D=F G=4$ and $B F=\\frac{1}{2}$, fin... | {
"ground_truth": "\\frac{16}{5}",
"style": "rule"
} | {
"index": "39985",
"split": "train"
} |
math | [
{
"content": "In the convex quadrilateral \\(ABCD\\), the length of side \\(AD\\) is 4, the length of side \\(CD\\) is 7, the cosine of angle \\(ADC\\) is \\(\\frac{1}{2}\\), and the sine of angle \\(BCA\\) is \\(\\frac{1}{3}\\). Find the length of side \\(BC\\) given that the circumcircle of triangle \\(ABC\\)... | {
"ground_truth": "\\frac{\\sqrt{37}}{3\\sqrt{3}}(\\sqrt{24} - 1)",
"style": "rule"
} | {
"index": "39986",
"split": "train"
} |
math | [
{
"content": "Suppose $f(x)$ and $g(x)$ are functions satisfying $f(g(x)) = x^2$ and $g(f(x)) = x^4$ for all $x \\ge 1.$ If $g(81) = 81,$ compute $[g(9)]^4.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "81",
"style": "rule"
} | {
"index": "39987",
"split": "train"
} |
math | [
{
"content": "The longer leg of a right triangle is $1$ foot shorter than twice the length of the shorter leg. The area of the triangle is $60$ square feet. What is the length of the hypotenuse, in feet? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "39988",
"split": "train"
} |
math | [
{
"content": "The pairwise products $a b, b c, c d$, and $d a$ of positive integers $a, b, c$, and $d$ are $64,88,120$, and 165 in some order. Find $a+b+c+d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "39989",
"split": "train"
} |
math | [
{
"content": "Kelvin the Frog is playing the game of Survival. He starts with two fair coins. Every minute, he flips all his coins one by one, and throws a coin away if it shows tails. The game ends when he has no coins left, and Kelvin's score is the *square* of the number of minutes elapsed. What is the expec... | {
"ground_truth": "\\frac{64}{9}",
"style": "rule"
} | {
"index": "39990",
"split": "train"
} |
math | [
{
"content": "What fraction of the large $12$ by $18$ rectangular region is shaded? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "39991",
"split": "train"
} |
math | [
{
"content": "On the lateral side $CD$ of trapezoid $ABCD (AD \\parallel BC)$, a point $M$ is marked. From vertex $A$, a perpendicular $AH$ is dropped to the segment $BM$. It is found that $AD = HD$. Find the length of segment $AD$ if it is known that $BC = 16$, $CM = 8$, $MD = 9$. Please output the final answe... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "39992",
"split": "train"
} |
math | [
{
"content": "How many ordered triples of integers $(a,b,c)$, with $a \\ge 2$, $b\\ge 1$, and $c \\ge 0$, satisfy both $\\log_a b = c^{2005}$ and $a + b + c = 2005$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39993",
"split": "train"
} |
math | [
{
"content": "For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\\frac{7}{51}$ is $0.\\overline{23}_k = 0.232323..._k$. What is $k$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "39994",
"split": "train"
} |
math | [
{
"content": "Let the set \\( T = \\{0, 1, \\dots, 6\\} \\),\n$$\nM = \\left\\{\\left.\\frac{a_1}{7}+\\frac{a_2}{7^2}+\\frac{a_3}{7^3}+\\frac{a_4}{7^4} \\right\\rvert\\, a_i \\in T, i=1,2,3,4\\right\\}.\n$$\nIf the elements of the set \\( M \\) are arranged in decreasing order, what is the 2015th number? Please... | {
"ground_truth": "\\frac{386}{2401}",
"style": "rule"
} | {
"index": "39995",
"split": "train"
} |
math | [
{
"content": "To meet market demand, a clothing supermarket purchased a short-sleeved $T$-shirt at the beginning of June. The cost price of each shirt is $80$ yuan, and the supermarket stipulates that the selling price of each shirt must not be less than $90$ yuan. According to a survey, when the selling price ... | {
"ground_truth": "12000",
"style": "rule"
} | {
"index": "39996",
"split": "train"
} |
math | [
{
"content": "A workshop produces products of types $A$ and $B$. Producing one unit of product $A$ requires 10 kg of steel and 23 kg of non-ferrous metals, while producing one unit of product $B$ requires 70 kg of steel and 40 kg of non-ferrous metals. The profit from selling one unit of product $A$ is 80 thous... | {
"ground_truth": "2180",
"style": "rule"
} | {
"index": "39997",
"split": "train"
} |
math | [
{
"content": "Find the largest natural number in which all digits are different and each pair of adjacent digits differs by 6 or 7. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60718293",
"style": "rule"
} | {
"index": "39998",
"split": "train"
} |
math | [
{
"content": "A can of soup can feed $3$ adults or $5$ children. If there are $5$ cans of soup and $15$ children are fed, then how many adults would the remaining soup feed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "39999",
"split": "train"
} |
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