data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In right triangle $ABC$ the hypotenuse $\\overline{AB}=5$ and leg $\\overline{AC}=3$. The bisector of angle $A$ meets the opposite side in $A_1$. A second right triangle $PQR$ is then constructed with hypotenuse $\\overline{PQ}=A_1B$ and leg $\\overline{PR}=A_1C$. If the bisector of angle $P$ meet... | {
"ground_truth": "\\frac{3\\sqrt{5}}{4}",
"style": "rule"
} | {
"index": "31100",
"split": "train"
} |
math | [
{
"content": "Find the integer \\(n\\), such that \\(-180 < n < 180\\), for which \\(\\tan n^\\circ = \\tan 276^\\circ.\\) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "31101",
"split": "train"
} |
math | [
{
"content": "Assume that the scores $X$ of 400,000 students in a math mock exam in Yunnan Province approximately follow a normal distribution $N(98,100)$. It is known that a student's score ranks among the top 9100 in the province. Then, the student's math score will not be less than ______ points. (Reference ... | {
"ground_truth": "118",
"style": "rule"
} | {
"index": "31102",
"split": "train"
} |
math | [
{
"content": "Given \\( a, b, c \\geq 0 \\), \\( t \\geq 1 \\), and satisfying\n\\[\n\\begin{cases}\na + b + c = \\frac{1}{2}, \\\\\n\\sqrt{a + \\frac{1}{2}(b - c)^{2}} + \\sqrt{b} + \\sqrt{c} = \\frac{\\sqrt{6t}}{2},\n\\end{cases}\n\\]\n\nfind \\( a^{2t} + b^{2t} + c^{2t} \\). Please output the final answer wi... | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "31103",
"split": "train"
} |
math | [
{
"content": "Simplify the expression $\\dfrac{\\sin(2\\pi-\\alpha)\\cos(\\pi+\\alpha)\\cos(\\frac{\\pi}{2}+\\alpha)\\cos(\\frac{11\\pi}{2}-\\alpha)}{\\cos(\\pi-\\alpha)\\sin(3\\pi-\\alpha)\\sin(-\\pi-\\alpha)\\sin(\\frac{9\\pi}{2}+\\alpha)\\tan(\\pi+\\alpha)}$. Please output the final answer within \\boxed{}."... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "31104",
"split": "train"
} |
math | [
{
"content": "Let $\\overrightarrow{{e_1}}$ and $\\overrightarrow{{e_2}}$ be non-collinear vectors. If $k\\overrightarrow{{e_1}}+4\\overrightarrow{{e_2}}$ and $\\overrightarrow{{e_1}}+k\\overrightarrow{{e_2}}$ are collinear and have opposite directions, then the value of $k$ is ____. Please output the final ans... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "31105",
"split": "train"
} |
math | [
{
"content": "A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$ , $BC = 12$ , $CA = 13$ . The probability that a circle with radius $\\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\\frac{m}{n}$ where $m... | {
"ground_truth": "61",
"style": "rule"
} | {
"index": "31106",
"split": "train"
} |
math | [
{
"content": "In the sequence ${a_{n}}$, $a_{n+1}=\\begin{cases} 2a_{n}\\left(a_{n} < \\frac{1}{2}\\right) \\\\ 2a_{n}-1\\left(a_{n}\\geqslant \\frac{1}{2}\\right) \\end{cases}$, if $a_{1}=\\frac{4}{5}$, then the value of $a_{20}$ is $\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "31107",
"split": "train"
} |
math | [
{
"content": "Calculate the sum of the square of the binomial coefficients: $C_2^2+C_3^2+C_4^2+…+C_{11}^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "220",
"style": "rule"
} | {
"index": "31108",
"split": "train"
} |
math | [
{
"content": "Let $T$ denote the value of the sum\\[\\sum_{n=0}^{432} (-1)^{n} {1500 \\choose 3n}\\]Determine the remainder obtained when $T$ is divided by $100$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "31109",
"split": "train"
} |
math | [
{
"content": "Given that $a-b=3$, find the value of $1+2b-(a+b)$.\nGiven that $2^x=3$, find the value of $2^{2x-3}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{8}",
"style": "rule"
} | {
"index": "31110",
"split": "train"
} |
math | [
{
"content": "In triangle $XYZ$, $XY=12$, $YZ=16$, and $XZ=20$. Point $M$ is on $\\overline{XY}$, $N$ is on $\\overline{YZ}$, and $O$ is on $\\overline{XZ}$. Let $XM = p \\cdot XY$, $YN = q \\cdot YZ$, and $ZO = r \\cdot XZ$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=3/4$ and $p^2+q^2+r^2=1/2$. Th... | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "31111",
"split": "train"
} |
math | [
{
"content": "Given a circle C with its center C on the positive x-axis and a radius of 5, the chord intercepted by the line $x-y+3=0$ has a length of $2\\sqrt{17}$.\n(1) Find the equation of circle C;\n(2) Suppose the line $ax-y+5=0$ intersects circle C at points A and B, find the range of the real number $a$;... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "31112",
"split": "train"
} |
math | [
{
"content": "Shift the graph of the function $f(x) = 2\\sin(2x + \\frac{\\pi}{4})$ to the right by $\\varphi (\\varphi > 0)$ units, then shrink the x-coordinate of each point on the graph to half of its original value (the y-coordinate remains unchanged), and make the resulting graph symmetric about the line $... | {
"ground_truth": "\\frac{3}{8}\\pi",
"style": "rule"
} | {
"index": "31113",
"split": "train"
} |
math | [
{
"content": "Given the equation \\(x^2 + y^2 = 2(|x| + |y|)\\), calculate the area of the region enclosed by its graph. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\pi",
"style": "rule"
} | {
"index": "31114",
"split": "train"
} |
math | [
{
"content": "What is the area of the circle defined by $x^2-6x +y^2-14y +33=0$ that lies beneath the line $y=7$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{25\\pi}{2}",
"style": "rule"
} | {
"index": "31115",
"split": "train"
} |
math | [
{
"content": "Evaluate: $5-7\\left(8-3^2\\right)4.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "31116",
"split": "train"
} |
math | [
{
"content": "Tom adds up all the even integers from 2 to 600, inclusive. Lara adds up all the integers from 1 to 200, inclusive. What is Tom's sum divided by Lara's sum? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "31117",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a_n\\}$, the sum of the first $n$ terms is $S_n$, and it is known that $a_2=1$, $S_4=8$. Find $a_5$ and $S_{10}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "31118",
"split": "train"
} |
math | [
{
"content": "Given a set of sample data with $8$ numbers, the average is $8$, and the variance is $12$. Two unknown numbers are added to this set of sample data to form a new set of sample data. It is known that the average of the new sample data is $9$. Find the minimum value of the variance of the new sample... | {
"ground_truth": "13.6",
"style": "rule"
} | {
"index": "31119",
"split": "train"
} |
math | [
{
"content": "On a line passing through the center $O$ of a circle with radius 12, points $A$ and $B$ are chosen such that $OA=15$, $AB=5$, and $A$ lies between $O$ and $B$. Tangents are drawn from points $A$ and $B$ to the circle, with the points of tangency lying on the same side of the line $OB$. Find the ar... | {
"ground_truth": "150/7",
"style": "rule"
} | {
"index": "31120",
"split": "train"
} |
math | [
{
"content": "Factor the following expression: $37a^2 +111a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37a(a+3)",
"style": "rule"
} | {
"index": "31121",
"split": "train"
} |
math | [
{
"content": "What is the smallest natural number that can be added to 40,317 to make it a palindrome? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "87",
"style": "rule"
} | {
"index": "31122",
"split": "train"
} |
math | [
{
"content": "$3^3 + 3^3 + 3^3 =$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$3^4$",
"style": "rule"
} | {
"index": "31123",
"split": "train"
} |
math | [
{
"content": "Let the bisectors of the exterior angles at $B$ and $C$ of triangle $ABC$ meet at $D$. Then, if all measurements are in degrees, angle $BDC$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}(180-A)",
"style": "rule"
} | {
"index": "31124",
"split": "train"
} |
math | [
{
"content": "Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{24}",
"style": "rule"
} | {
"index": "31125",
"split": "train"
} |
math | [
{
"content": "Find the number of degrees in the measure of angle $x$.\n\n[asy]\nimport markers;\nsize (5cm,5cm);\npair A,B,C,D,F,H;\n\nA=(0,0);\nB=(5,0);\nC=(9,0);\nD=(3.8,7);\n\nF=(2.3,7.2);\nH=(5.3,7.2);\n\ndraw((4.2,6.1){up}..{right}(5.3,7.2));\ndraw((3.6,6.1){up}..{left}(2.3,7.2));\n\ndraw (A--B--C--D--A);\... | {
"ground_truth": "82^\\circ",
"style": "rule"
} | {
"index": "31126",
"split": "train"
} |
math | [
{
"content": "For integers $a, b, c, d$, let $f(a, b, c, d)$ denote the number of ordered pairs of integers $(x, y) \\in \\{1,2,3,4,5\\}^{2}$ such that $a x+b y$ and $c x+d y$ are both divisible by 5. Find the sum of all possible values of $f(a, b, c, d)$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "31127",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), points \\(P\\) and \\(Q\\) are taken on the base \\(AC\\) such that \\(AP < AQ\\). The lines \\(BP\\) and \\(BQ\\) divide the median \\(AM\\) into three equal parts. It is known that \\(PQ = 3\\).\n\nFind \\(AC\\). Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31128",
"split": "train"
} |
math | [
{
"content": "There are $2022$ equally spaced points on a circular track $\\gamma$ of circumference $2022$. The points are labeled $A_1, A_2, \\ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3... | {
"ground_truth": "2042222",
"style": "rule"
} | {
"index": "31129",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\frac{3+x(3+x)-3^2}{x-3+x^2}$ for $x=-2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "31130",
"split": "train"
} |
math | [
{
"content": "In the eight-term sequence $A,B,C,D,E,F,G,H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. What is $A+H$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "31131",
"split": "train"
} |
math | [
{
"content": "Arrange positive integers that are neither perfect squares nor perfect cubes (excluding 0) in ascending order as 2, 3, 5, 6, 7, 10, ..., and determine the 1000th number in this sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1039",
"style": "rule"
} | {
"index": "31132",
"split": "train"
} |
math | [
{
"content": "An eight-digit integer is formed by repeating a positive four-digit integer. For example, 25,632,563 or 60,786,078 are integers of this form. What is the greatest common divisor of all eight-digit integers of this form? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10001",
"style": "rule"
} | {
"index": "31133",
"split": "train"
} |
math | [
{
"content": "Solve in prime numbers the equation $x^y - y^x = xy^2 - 19$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(2, 3)(2, 7)",
"style": "rule"
} | {
"index": "31134",
"split": "train"
} |
math | [
{
"content": "Calculate the corrected average score and variance of a class of 50 students, given that the original average was 70 and the original variance was 102, after two students' scores were corrected from 50 to 80 and from 90 to 60. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "31135",
"split": "train"
} |
math | [
{
"content": "Given the polar equation of curve $C_1$ is $\\rho^2=\\frac {2}{3+\\cos2\\theta}$, establish a rectangular coordinate system with the pole O as the origin and the polar axis as the positive direction of the x-axis. After stretching all the x-coordinates of points on curve $C_1$ to twice their origi... | {
"ground_truth": "\\frac{13\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "31136",
"split": "train"
} |
math | [
{
"content": "For how many positive integers $k$ do the lines with equations $9x+4y=600$ and $kx-4y=24$ intersect at a point whose coordinates are positive integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "31137",
"split": "train"
} |
math | [
{
"content": "An isosceles triangle has side lengths 8 cm, 8 cm and 10 cm. The longest side of a similar triangle is 25 cm. What is the perimeter of the larger triangle, in centimeters? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "65",
"style": "rule"
} | {
"index": "31138",
"split": "train"
} |
math | [
{
"content": "In triangle $XYZ$, $\\angle Z=90^\\circ$, $XZ=3$ and $YZ=4$. Points $W$ and $V$ are on $\\overline{XY}$ and $\\overline{YZ}$, respectively, and $\\angle WVZ=90^\\circ$. If $WV=2$, then what is the length of $WY$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10}{3}",
"style": "rule"
} | {
"index": "31139",
"split": "train"
} |
math | [
{
"content": "At breakfast, lunch, and dinner, Joe randomly chooses with equal probabilities either an apple, an orange, or a banana to eat. On a given day, what is the probability that Joe will eat at least two different kinds of fruit? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "\\frac{8}{9}",
"style": "rule"
} | {
"index": "31140",
"split": "train"
} |
math | [
{
"content": "Aerith timed herself solving a contest and noted the time both as days:hours:minutes:seconds and in seconds. For example, if she spent 1,000,000 seconds, she recorded it as 11:13:46:40 and 1,000,000 seconds. Bob subtracts these numbers, ignoring punctuation. In this case, he computes:\n\n\\[ 11134... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "31141",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $C$: $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) with asymptotic equations $y = \\pm \\sqrt{3}x$, and $O$ as the origin, the point $M(\\sqrt{5}, \\sqrt{3})$ lies on the hyperbola.\n\n$(1)$ Find the equation of the hyperbola $C$.\n\n$(2)$ If a line $l$ intersects ... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "31142",
"split": "train"
} |
math | [
{
"content": "Given positive real numbers \\(a, b, c\\) satisfy \\(2(a+b)=ab\\) and \\(a+b+c=abc\\), find the maximum value of \\(c\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{15}",
"style": "rule"
} | {
"index": "31143",
"split": "train"
} |
math | [
{
"content": "Expanding $(1+0.2)^{1000}_{}$ by the binomial theorem and doing no further manipulation gives\n${1000 \\choose 0}(0.2)^0+{1000 \\choose 1}(0.2)^1+{1000 \\choose 2}(0.2)^2+\\cdots+{1000 \\choose 1000}(0.2)^{1000}$ $= A_0 + A_1 + A_2 + \\cdots + A_{1000},$ where $A_k = {1000 \\choose k}(0.2)^k$ for ... | {
"ground_truth": "166",
"style": "rule"
} | {
"index": "31144",
"split": "train"
} |
math | [
{
"content": "Find the matrix that corresponds to projecting onto the vector $\\begin{pmatrix} 2 \\\\ -3 \\end{pmatrix}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} 4/13 & -6/13 \\\\ -6/13 & 9/13 \\end{pmatrix}",
"style": "rule"
} | {
"index": "31145",
"split": "train"
} |
math | [
{
"content": "How many ways are there to color every integer either red or blue such that \\(n\\) and \\(n+7\\) are the same color for all integers \\(n\\), and there does not exist an integer \\(k\\) such that \\(k, k+1\\), and \\(2k\\) are all the same color? Please output the final answer within \\boxed{}.",... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "31146",
"split": "train"
} |
math | [
{
"content": "Eight hockey teams are competing against each other in a single round to advance to the semifinals. What is the minimum number of points that guarantees a team advances to the semifinals? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31147",
"split": "train"
} |
math | [
{
"content": "Remove all perfect squares from the sequence of positive integers \\(1, 2, 3, \\cdots\\) to get a new sequence, and calculate the 2003rd term of this new sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2047",
"style": "rule"
} | {
"index": "31148",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a trapezium with $AB// DC, AB = b, AD = a ,a<b$ and $O$ the intersection point of the diagonals. Let $S$ be the area of the trapezium $ABCD$ . Suppose the area of $\\vartriangle DOC$ is $2S/9$ . Find the value of $a/b$ . Please output the final answer within \\boxed{}.",... | {
"ground_truth": "\\frac{2 + 3\\sqrt{2}}{7}",
"style": "rule"
} | {
"index": "31149",
"split": "train"
} |
math | [
{
"content": "A regular dodecahedron is projected orthogonally onto a plane, and its image is an $n$-sided polygon. What is the smallest possible value of $n$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "31150",
"split": "train"
} |
math | [
{
"content": "What percent of the positive integers less than or equal to $100$ have no remainders when divided by $5?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "31151",
"split": "train"
} |
math | [
{
"content": "Given the parabola $y^{2}=4x$, and the line $l$: $y=- \\frac {1}{2}x+b$ intersects the parabola at points $A$ and $B$.\n(I) If the $x$-axis is tangent to the circle with $AB$ as its diameter, find the equation of the circle;\n(II) If the line $l$ intersects the negative semi-axis of $y$, find the ... | {
"ground_truth": "\\frac {32 \\sqrt {3}}{9}",
"style": "rule"
} | {
"index": "31152",
"split": "train"
} |
math | [
{
"content": "Calculate the monotonic intervals of $F(x)=\\int_{0}^{x}{(t^{2}+2t-8)dt}$ for $x > 0$.\n\n(1) Determine the monotonic intervals of $F(x)$.\n\n(2) Find the maximum and minimum values of the function $F(x)$ on the interval $[1,2]$. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "-\\frac{28}{3}",
"style": "rule"
} | {
"index": "31153",
"split": "train"
} |
math | [
{
"content": "If \\( a \\) and \\( b \\) are positive numbers such that \\( a^b = b^a \\) and \\( b = 4a \\), then find the value of \\( a \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt[3]{4}",
"style": "rule"
} | {
"index": "31154",
"split": "train"
} |
math | [
{
"content": "Let $d_1 = a^2 + 3^a + a \\cdot 3^{(a+1)/2}$ and $d_2 = a^2 + 3^a - a \\cdot 3^{(a+1)/2}$. If $1 \\le a \\le 300$, how many integral values of $a$ are there such that $d_1 \\cdot d_2$ is a multiple of $7$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "43",
"style": "rule"
} | {
"index": "31155",
"split": "train"
} |
math | [
{
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are unit vectors, and the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is $90^{\\circ}$, if vector $\\overrightarrow{c}$ satisfies $|\\overrightarrow{c}- \\overrightarrow{a} \\overrightarrow{b}|=2$, calculate the maximum v... | {
"ground_truth": "2 + \\sqrt{2}",
"style": "rule"
} | {
"index": "31156",
"split": "train"
} |
math | [
{
"content": "Given that the domains of functions f(x) and g(x) are both R, and f(x) + g(2-x) = 5, g(x) - f(x-4) = 7. If the graph of y=g(x) is symmetric about the line x=2, g(2) = 4, find the sum of the values of f(k) for k from 1 to 22. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "-24",
"style": "rule"
} | {
"index": "31157",
"split": "train"
} |
math | [
{
"content": "Trapezoid $ABCD$ has base $AB = 20$ units and base $CD = 30$ units. Diagonals $AC$ and $BD$ intersect at $X$. If the area of trapezoid $ABCD$ is $300$ square units, what is the area of triangle $BXC$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "31158",
"split": "train"
} |
math | [
{
"content": "Given $f(\\alpha)= \\frac{\\sin (\\alpha-3\\pi)\\cos (2\\pi-\\alpha)\\cdot\\sin (-\\alpha+ \\frac{3}{2}\\pi)}{\\cos (-\\pi-\\alpha)\\sin (-\\pi-\\alpha)}$. \n$(1)$ Simplify $f(\\alpha)$. \n$(2)$ If $\\alpha$ is an angle in the third quadrant, and $\\cos \\left(\\alpha- \\frac{3}{2}\\pi\\right)= ... | {
"ground_truth": "\\frac{2\\sqrt{6}}{5}",
"style": "rule"
} | {
"index": "31159",
"split": "train"
} |
math | [
{
"content": "On the Cartesian plane, find the number of integer coordinate points (points where both x and y are integers) that satisfy the following system of inequalities:\n\\[\n\\begin{cases}\ny \\leq 3x, \\\\\ny \\geq \\frac{1}{3}x, \\\\\nx + y \\leq 100.\n\\end{cases}\n\\] Please output the final answer w... | {
"ground_truth": "2551",
"style": "rule"
} | {
"index": "31160",
"split": "train"
} |
math | [
{
"content": "In a quadrilateral $WXYZ$, the angles satisfy $\\angle W = 3\\angle X = 4\\angle Y = 6\\angle Z$. Determine the exact degree measure of $\\angle W$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1440}{7}",
"style": "rule"
} | {
"index": "31161",
"split": "train"
} |
math | [
{
"content": "An archipelago consists of \\( N \\geq 7 \\) islands. Any two islands are connected by no more than one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are necessarily two that are connected by a bridge. What is the maximum value that \\( N \\)... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "31162",
"split": "train"
} |
math | [
{
"content": "For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $500$ that are neither $3$-nice nor $5$-nice. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "266",
"style": "rule"
} | {
"index": "31163",
"split": "train"
} |
math | [
{
"content": "Given that in a class test, $15\\%$ of the students scored $60$ points, $50\\%$ scored $75$ points, $20\\%$ scored $85$ points, and the rest scored $95$ points, calculate the difference between the mean and median score of the students' scores on this test. Please output the final answer within \\... | {
"ground_truth": "2.75",
"style": "rule"
} | {
"index": "31164",
"split": "train"
} |
math | [
{
"content": "In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\\triangle ABC$ is $50$ units and the area of $\\triangle ABC$ is $100$ square units? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "31165",
"split": "train"
} |
math | [
{
"content": "Teacher Xixi and teacher Shanshan are teachers in the senior and junior classes of a kindergarten, respectively. Teacher Xixi prepared a large bag of apples to distribute to her students, giving exactly 3 apples to each child; teacher Shanshan prepared a large bag of oranges to distribute to her s... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "31166",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system xOy, there is a line l₁: x = 2, and a curve C: {x = 2cosϕ, y = 2 + 2sinϕ} (where ϕ is a parameter). With O as the pole and the non-negative half-axis of the x-axis as the polar axis, establish a polar coordinate system. The polar coordinates of point M are $(3,... | {
"ground_truth": "\\frac {3 \\sqrt {3}}{2}",
"style": "rule"
} | {
"index": "31167",
"split": "train"
} |
math | [
{
"content": "Find the number of different arrangements for a class to select 6 people to participate in two volunteer activities, with each activity accommodating no more than 4 people. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "31168",
"split": "train"
} |
math | [
{
"content": "If I have a $5\\times5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and each row of the board contains no more than one pawn? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14400",
"style": "rule"
} | {
"index": "31169",
"split": "train"
} |
math | [
{
"content": "In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. Given that $(a+c)^2 = b^2 + 2\\sqrt{3}ac\\sin C$.\n1. Find the measure of angle B.\n2. If $b=8$, $a>c$, and the area of triangle ABC is $3\\sqrt{3}$, find the value of $a$. Please output the final answer within... | {
"ground_truth": "5 + \\sqrt{13}",
"style": "rule"
} | {
"index": "31170",
"split": "train"
} |
math | [
{
"content": "Let N = $69^5 + 5 \\cdot 69^4 + 10 \\cdot 69^3 + 10 \\cdot 69^2 + 5 \\cdot 69 + 1$. How many positive integers are factors of $N$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "216",
"style": "rule"
} | {
"index": "31171",
"split": "train"
} |
math | [
{
"content": "Let \\(ABCD\\) be a rectangle, and let \\(E\\) and \\(F\\) be points on segment \\(AB\\) such that \\(AE = EF = FB\\). If \\(CE\\) intersects the line \\(AD\\) at \\(P\\), and \\(PF\\) intersects \\(BC\\) at \\(Q\\), determine the ratio of \\(BQ\\) to \\(CQ\\). Please output the final answer withi... | {
"ground_truth": "1/3",
"style": "rule"
} | {
"index": "31172",
"split": "train"
} |
math | [
{
"content": "If: (1) \\( a, b, c, d \\) are all elements of \\( \\{1,2,3,4\\} \\); (2) \\( a \\neq b, b \\neq c, c \\neq d, d \\neq a \\); (3) \\( a \\) is the smallest value among \\( a, b, c, d \\),\n\nthen, how many different four-digit numbers \\( \\overline{abcd} \\) can be formed? Please output the final... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "31173",
"split": "train"
} |
math | [
{
"content": "Find the largest constant $C$ so that for all real numbers $x$, $y$, and $z$,\n\\[x^2 + y^2 + z^3 + 1 \\ge C(x + y + z).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "31174",
"split": "train"
} |
math | [
{
"content": "Given that the cosine value of the vertex angle of an isosceles triangle equals $\\dfrac{4}{5}$, calculate the sine value of the base angle of this triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{2\\sqrt{3}}{5}",
"style": "rule"
} | {
"index": "31175",
"split": "train"
} |
math | [
{
"content": "Let $S_{n}$ be the sum of the first $n$ terms of a geometric sequence $\\{a_{n}\\}$. If $S_{4}=-5$ and $S_{6}=21S_{2}$, calculate the value of $S_{8}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-85",
"style": "rule"
} | {
"index": "31176",
"split": "train"
} |
math | [
{
"content": "A certain school club has 10 members, and two of them are put on duty each day from Monday to Friday. Given that members A and B must be scheduled on the same day, and members C and D cannot be scheduled together, the total number of different possible schedules is (▲). Choices:\n\nA) 21600\n\nB) ... | {
"ground_truth": "5400",
"style": "rule"
} | {
"index": "31177",
"split": "train"
} |
math | [
{
"content": "Given \\\\((a+b-c)(a+b+c)=3ab\\\\) and \\\\(c=4\\\\), the maximum area of \\\\(\\Delta ABC\\\\) is \\_\\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "31178",
"split": "train"
} |
math | [
{
"content": "Let $A = \\{1, 2, 3, 4, 5, 6, 7\\}$, and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "399",
"style": "rule"
} | {
"index": "31179",
"split": "train"
} |
math | [
{
"content": "Medians $\\overline{DP}$ and $\\overline{EQ}$ of $\\triangle DEF$ are perpendicular. If $DP= 18$ and $EQ = 24$, then what is ${DF}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8\\sqrt{13}",
"style": "rule"
} | {
"index": "31180",
"split": "train"
} |
math | [
{
"content": "In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?\n[... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "31181",
"split": "train"
} |
math | [
{
"content": "The minimum positive period and maximum value of the function $f\\left(x\\right)=\\sin \\frac{x}{3}+\\cos \\frac{x}{3}$ are respectively $3\\pi$ and $\\sqrt{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "31182",
"split": "train"
} |
math | [
{
"content": "The product of two 2-digit numbers is $5488$. What is the smaller of the two numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "31183",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given vectors $\\overrightarrow{m} = (\\cos A, \\sin A)$ and $\\overrightarrow{n} = (\\cos B, -\\sin B)$, and $|\\overrightarrow{m} - \\overrightarrow{n}| = 1$.\n\n(1) Find the degree measure of a... | {
"ground_truth": "\\frac{3\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "31184",
"split": "train"
} |
math | [
{
"content": "$\\log p+\\log q=\\log(p+q)$ only if: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "p=\\frac{q}{q-1}",
"style": "rule"
} | {
"index": "31185",
"split": "train"
} |
math | [
{
"content": "Given the word 'ARROW', find the probability that a random arrangement of its letters will have both R's next to each other. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "31186",
"split": "train"
} |
math | [
{
"content": "The product of the roots of the equation \\((x-4)(x-2)+(x-2)(x-6)=0\\) is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31187",
"split": "train"
} |
math | [
{
"content": "For what value of $x$ does $3^{3x^{2} - 8x + 5} = 3^{3x^{2} + 5x - 6}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{13}",
"style": "rule"
} | {
"index": "31188",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $C: \\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a>b>0)$, its foci are equal to the minor axis length of the ellipse $Ω:x^{2}+ \\frac{y^{2}}{4}=1$, and the major axis lengths of C and Ω are equal.\n(1) Find the equation of ellipse C;\n(2) Let $F_1$, $F_2$ be the left and right fo... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "31189",
"split": "train"
} |
math | [
{
"content": "You flip a fair coin which results in heads ( $\\text{H}$ ) or tails ( $\\text{T}$ ) with equal probability. What is the probability that you see the consecutive sequence $\\text{THH}$ before the sequence $\\text{HHH}$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "\\frac{7}{8}",
"style": "rule"
} | {
"index": "31190",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the internal angles $A$, $B$, and $C$ satisfy the equation $$2(\\tan B + \\tan C) = \\frac{\\tan B}{\\cos C} + \\frac{\\tan C}{\\cos B}$$. Find the minimum value of $\\cos A$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31191",
"split": "train"
} |
math | [
{
"content": "Circles $\\mathcal{C}_{1}$ and $\\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6)$, and the product of the radii is $68$. The x-axis and the line $y = mx$, where $m > 0$, are tangent to both circles. It is given that $m$ can be written in the form $a\\sqrt {b}/c$, where $a$, $b$, a... | {
"ground_truth": "282",
"style": "rule"
} | {
"index": "31192",
"split": "train"
} |
math | [
{
"content": "The four hydrogen atoms in the methane molecule $\\mathrm{CH}_{4}$ are located at the vertices of a regular tetrahedron with edge length 1. The carbon atom $C$ is located at the center of the tetrahedron $C_{0}$. Let the four hydrogen atoms be $H_{1}, H_{2}, H_{3}, H_{4}$. Then $\\sum_{1 \\leq i <... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "31193",
"split": "train"
} |
math | [
{
"content": "How many zeros are at the end of the product $s(1) \\cdot s(2) \\cdot \\ldots \\cdot s(100)$, where $s(n)$ denotes the sum of the digits of the natural number $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "31194",
"split": "train"
} |
math | [
{
"content": "Given that a school has 5 top students and 3 teachers, where each teacher mentors no more than 2 students, calculate the number of different mentorship arrangements possible. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "31195",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $a=1$, $B=45^{\\circ}$, $S_{\\triangle ABC}=2$, find the diameter of the circumcircle of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5 \\sqrt {2}",
"style": "rule"
} | {
"index": "31196",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A, B, C$ are denoted as $a, b, c$, respectively, and $a=1, A=\\frac{\\pi}{6}$.\n\n(Ⅰ) When $b=\\sqrt{3}$, find the magnitude of angle $C$; \n(Ⅱ) Find the maximum area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "\\frac{2+ \\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "31197",
"split": "train"
} |
math | [
{
"content": "A company buys an assortment of 150 pens from a catalog for \\$15.00. Shipping costs an additional \\$5.50. Furthermore, they receive a 10% discount on the total price due to a special promotion. What is the average cost, in cents, for each pen? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "31198",
"split": "train"
} |
math | [
{
"content": "In rectangle \\(ABCD\\), \\(\\overline{AB}=30\\) and \\(\\overline{BC}=15\\). Let \\(E\\) be a point on \\(\\overline{CD}\\) such that \\(\\angle CBE=45^\\circ\\) and \\(\\triangle ABE\\) is isosceles. Find \\(\\overline{AE}.\\) Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "31199",
"split": "train"
} |
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