data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given a general triangle \\(ABC\\) with points \\(K, L, M, N, U\\) on its sides:\n\n- Point \\(K\\) is the midpoint of side \\(AC\\).\n- Point \\(U\\) is the midpoint of side \\(BC\\).\n- Points \\(L\\) and \\(M\\) lie on segments \\(CK\\) and \\(CU\\) respectively, such that \\(LM \\parallel KU\\... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "8200",
"split": "train"
} |
math | [
{
"content": "Let $I, T, E, S$ be distinct positive integers such that the product $ITEST = 2006$ . What is the largest possible value of the sum $I + T + E + S + T + 2006$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2086",
"style": "rule"
} | {
"index": "8201",
"split": "train"
} |
math | [
{
"content": "Oleg drew an empty $50 \\times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrote the sum of the numbers writ... | {
"ground_truth": "1250",
"style": "rule"
} | {
"index": "8202",
"split": "train"
} |
math | [
{
"content": "The sum of the absolute values of the terms of a finite arithmetic progression is equal to 100. If all its terms are increased by 1 or all its terms are increased by 2, in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to 100. What values can... | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "8203",
"split": "train"
} |
math | [
{
"content": "A point $P$ is randomly placed in the interior of the right triangle below. What is the probability that the area of triangle $PBC$ is less than half of the area of triangle $ABC$? Express your answer as a common fraction. [asy]\nsize(7cm);\ndefaultpen(linewidth(0.7));\npair A=(0,5), B=(8,0), C=... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "8204",
"split": "train"
} |
math | [
{
"content": "Given a right circular cone with three mutually perpendicular side edges, each with a length of $\\sqrt{3}$, determine the surface area of the circumscribed sphere. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9\\pi",
"style": "rule"
} | {
"index": "8205",
"split": "train"
} |
math | [
{
"content": "In the diagram below, $\\overline{AB}\\parallel \\overline{CD}$ and $\\angle AXE$ is $108^\\circ$ less than 3 times $\\angle CYX$. Find $\\angle BXY$.\n\n[asy]\n\nunitsize(1inch);\n\npair A,B,C,D,X,Y,EE,F;\n\nA = (0,0);\n\nB=(1,0);\n\nC = (0,0.8);\n\nD=(1,0.8);\n\nEE = (0.35,-0.3);\n\nF = (0.8,1.... | {
"ground_truth": "54^\\circ",
"style": "rule"
} | {
"index": "8206",
"split": "train"
} |
math | [
{
"content": "$\\frac{x^{2}}{9} + \\frac{y^{2}}{7} = 1$, where $F_{1}$ and $F_{2}$ are the foci of the ellipse. Given that point $A$ lies on the ellipse and $\\angle AF_{1}F_{2} = 45^{\\circ}$, find the area of triangle $AF_{1}F_{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{2}",
"style": "rule"
} | {
"index": "8207",
"split": "train"
} |
math | [
{
"content": "The sum of the first 2011 terms of a geometric sequence is 200. The sum of the first 4022 terms is 380. Find the sum of the first 6033 terms. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "542",
"style": "rule"
} | {
"index": "8208",
"split": "train"
} |
math | [
{
"content": "Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indef... | {
"ground_truth": "5/11",
"style": "rule"
} | {
"index": "8209",
"split": "train"
} |
math | [
{
"content": "What is $(a^3+b^3)\\div(a^2-ab+b^2)$ when $a=5$ and $b=4$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "8210",
"split": "train"
} |
math | [
{
"content": "In a triangle, the lengths of the three sides are integers \\( l, m, n \\), with \\( l > m > n \\). It is known that \\( \\left\\{ \\frac{3^{l}}{10^{4}} \\right\\} = \\left\\{ \\frac{3^{m}}{10^{4}} \\right\\} = \\left\\{ \\frac{3^{n}}{10^{4}} \\right\\} \\), where \\( \\{x\\} \\) denotes the fract... | {
"ground_truth": "3003",
"style": "rule"
} | {
"index": "8211",
"split": "train"
} |
math | [
{
"content": "Square $ABCD$ has side length $30$. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$. The centroids of $\\triangle{ABP}$, $\\triangle{BCP}$, $\\triangle{CDP}$, and $\\triangle{DAP}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral?\n[asy] unitsize(... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "8212",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate plane $(xOy)$, two acute angles $\\alpha$ and $\\beta$ are formed with the non-negative semi-axis of $Ox$ as the initial side. Their terminal sides intersect the unit circle at points $A$ and $B$ respectively. The vertical coordinates of $A$ and $B$ are $\\frac{\\sqrt{5... | {
"ground_truth": "\\frac{3\\sqrt{10}}{10}",
"style": "rule"
} | {
"index": "8213",
"split": "train"
} |
math | [
{
"content": "A regular dodecagon \\( Q_1 Q_2 \\dotsb Q_{12} \\) is drawn in the coordinate plane with \\( Q_1 \\) at \\( (4,0) \\) and \\( Q_7 \\) at \\( (2,0) \\). If \\( Q_n \\) is the point \\( (x_n,y_n) \\), compute the numerical value of the product\n\\[\n(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \\dotsm (x... | {
"ground_truth": "531440",
"style": "rule"
} | {
"index": "8214",
"split": "train"
} |
math | [
{
"content": "Draw a rectangle. Connect the midpoints of the opposite sides to get 4 congruent rectangles. Connect the midpoints of the lower right rectangle for a total of 7 rectangles. Repeat this process infinitely. Let $n$ be the minimum number of colors we can assign to the rectangles so that no two rectan... | {
"ground_truth": "(3,4)",
"style": "rule"
} | {
"index": "8215",
"split": "train"
} |
math | [
{
"content": "In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and each head always tells the truth or always lies. Additionally, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:\n\n- 1st he... | {
"ground_truth": "176",
"style": "rule"
} | {
"index": "8216",
"split": "train"
} |
math | [
{
"content": "A $5 \\times 5$ square grid has the number -3 written in the upper-left square and the number 3 written in the lower-right square. In how many ways can the remaining squares be filled in with integers so that any two adjacent numbers differ by 1, where two squares are adjacent if they share a comm... | {
"ground_truth": "250",
"style": "rule"
} | {
"index": "8217",
"split": "train"
} |
math | [
{
"content": "$\\_$\\_$\\_$:20=24÷$\\_$\\_$\\_$=80%=$\\_$\\_$\\_$(fill in the blank with a fraction)=$\\_$\\_$\\_$(fill in the blank with a decimal) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.8",
"style": "rule"
} | {
"index": "8218",
"split": "train"
} |
math | [
{
"content": "Find the complex number $z$ such that\n\\[|z - 1| = |z + 3| = |z - i|.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1 - i",
"style": "rule"
} | {
"index": "8219",
"split": "train"
} |
math | [
{
"content": "Given that $θ∈[0,π]$, find the probability that $\\sin (θ+ \\frac {π}{3}) < \\frac {1}{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "8220",
"split": "train"
} |
math | [
{
"content": "Given that Jeff, Maria, and Lee paid $90, $150, and $210 respectively, find j - m where Jeff gave Lee $j dollars and Maria gave Lee $m dollars to settle the debts such that everyone paid equally. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "8221",
"split": "train"
} |
math | [
{
"content": "The segments of two lines, enclosed between two parallel planes, are in the ratio of \\( 5:9 \\), and the acute angles between these lines and one of the planes are in the ratio of \\( 2:1 \\), respectively. Find the cosine of the smaller angle. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "0.9",
"style": "rule"
} | {
"index": "8222",
"split": "train"
} |
math | [
{
"content": "Points $M$ , $N$ , $P$ are selected on sides $\\overline{AB}$ , $\\overline{AC}$ , $\\overline{BC}$ , respectively, of triangle $ABC$ . Find the area of triangle $MNP$ given that $AM=MB=BP=15$ and $AN=NC=CP=25$ .\n\n*Proposed by Evan Chen* Please output the final answer within \\boxe... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "8223",
"split": "train"
} |
math | [
{
"content": "Compute $\\arcsin (-1).$ Express your answer in radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "8224",
"split": "train"
} |
math | [
{
"content": "How many real numbers $x^{}_{}$ satisfy the equation $\\frac{1}{5}\\log_2 x = \\sin (5\\pi x)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "159",
"style": "rule"
} | {
"index": "8225",
"split": "train"
} |
math | [
{
"content": "Suppose $a$ and $b$ be positive integers not exceeding 100 such that $$a b=\\left(\\frac{\\operatorname{lcm}(a, b)}{\\operatorname{gcd}(a, b)}\\right)^{2}$$ Compute the largest possible value of $a+b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "8226",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)$ ($x \\in \\mathbb{R}$) that satisfies $f(x+\\pi)=f(x)+\\sin x$, and $f(x)=0$ when $0 \\leqslant x < \\pi$, determine the value of $f(\\frac{23\\pi}{6})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "8227",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\\\_a{n}\\_$, where $\\_a{n}>0$, $\\_a{1}=1$, and $\\_a{n+2}=\\frac{1}{a{n}+1}$, and it is known that $\\_a{6}=a{2}$, find the value of $\\_a{2016}+a{3}=\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "8228",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b=2$.\n\n(1) If angles $A$, $B$, $C$ form an arithmetic progression, find the radius of the circumcircle of $\\triangle ABC$.\n(2) If sides $a$, $b$, $c$ form an arithmetic progression, find the ma... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "8229",
"split": "train"
} |
math | [
{
"content": "Laura and her grandmother Ana just discovered that last year, their ages were divisible by 8 and that next year, their ages will be divisible by 7. Grandma Ana is not yet 100 years old. What is Laura's age? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "8230",
"split": "train"
} |
math | [
{
"content": "We measured the elevation angles of a tower standing on a horizontal plane from points $50 \\mathrm{~m}$ and $100 \\mathrm{~m}$ away from the base of the tower. The sum of the measured angles is $45^{\\circ}$. How tall is the tower? Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "28.08",
"style": "rule"
} | {
"index": "8231",
"split": "train"
} |
math | [
{
"content": "Regular decagon \\( ABCDEFGHIJ \\) has its center at \\( K \\). Each of the vertices and the center are to be associated with one of the digits \\( 1 \\) through \\( 10 \\), with each digit used exactly once, in such a way that the sums of the numbers on the lines \\( AKF \\), \\( BKG \\), \\( CKH... | {
"ground_truth": "3840",
"style": "rule"
} | {
"index": "8232",
"split": "train"
} |
math | [
{
"content": "Suppose that 9 boys and 15 girls line up in a row, but the arrangement must start with a boy and end with a girl. Let $T$ be the number of places in the row where a boy and a girl are standing next to each other. Calculate the average value of $T$ when all possible orders of these 24 people under ... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "8233",
"split": "train"
} |
math | [
{
"content": "Find all prime numbers $p$ for which there exists a unique $a \\in\\{1,2, \\ldots, p\\}$ such that $a^{3}-3 a+1$ is divisible by $p$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "8234",
"split": "train"
} |
math | [
{
"content": "Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \\times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be mark... | {
"ground_truth": "1007",
"style": "rule"
} | {
"index": "8235",
"split": "train"
} |
math | [
{
"content": "Complex numbers \\(a\\), \\(b\\), \\(c\\) form an equilateral triangle with side length 24 in the complex plane. If \\(|a + b + c| = 48\\), find \\(|ab + ac + bc|\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "768",
"style": "rule"
} | {
"index": "8236",
"split": "train"
} |
math | [
{
"content": "Given the equation about $x$, $2x^{2}-( \\sqrt {3}+1)x+m=0$, its two roots are $\\sin θ$ and $\\cos θ$, where $θ∈(0,π)$. Find:\n$(1)$ the value of $m$;\n$(2)$ the value of $\\frac {\\tan θ\\sin θ}{\\tan θ-1}+ \\frac {\\cos θ}{1-\\tan θ}$;\n$(3)$ the two roots of the equation and the value of $θ$ a... | {
"ground_truth": "\\frac {1}{2}",
"style": "rule"
} | {
"index": "8237",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than $101$ are multiples of either $5$ or $7$, but not both at once? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "8238",
"split": "train"
} |
math | [
{
"content": "Andy, Beth, Charlie, and Daniel take a test with thirty questions. Andy and Beth together get the same number of questions wrong as Charlie and Daniel together. Andy and Daniel together get four more questions wrong than Beth and Charlie do together. If Charlie gets five questions wrong, how many ... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "8239",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{v}$ be a vector such that\n\\[\\left\\| \\mathbf{v} + \\begin{pmatrix} 3 \\\\ -1 \\end{pmatrix} \\right\\| = 8.\\]Find the smallest possible value of $\\|\\mathbf{v}\\|.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8 - \\sqrt{10}",
"style": "rule"
} | {
"index": "8240",
"split": "train"
} |
math | [
{
"content": "A leak formed in the hold of a ship. A pump was immediately switched on to remove the water, but it couldn't keep up, and after 10 minutes, the water level rose by 20 cm. Then, a second pump of equal power was turned on, and after 5 minutes, the water level dropped by 10 cm. The leak was then seal... | {
"ground_truth": "1.25",
"style": "rule"
} | {
"index": "8241",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle A B C$ be a triangle with $A B=7, B C=1$, and $C A=4 \\sqrt{3}$. The angle trisectors of $C$ intersect $\\overline{A B}$ at $D$ and $E$, and lines $\\overline{A C}$ and $\\overline{B C}$ intersect the circumcircle of $\\triangle C D E$ again at $X$ and $Y$, respectively. Find the l... | {
"ground_truth": "\\frac{112}{65}",
"style": "rule"
} | {
"index": "8242",
"split": "train"
} |
math | [
{
"content": "Let\n\\[\\bold{A} = \\begin{pmatrix} 0 & 1 & 2 \\\\ 1 & 0 & 1 \\\\ 2 & 1 & 0 \\end{pmatrix}.\\]There exist constants $p$, $q$, and $r$ such that\n\\[\\bold{A}^3 + p \\bold{A}^2 + q \\bold{A} + r \\bold{I} = \\bold{0},\\]where $\\bold{I}$ and $\\bold{0}$ are the $3 \\times 3$ identity matrix and ze... | {
"ground_truth": "(0,-6,-4)",
"style": "rule"
} | {
"index": "8243",
"split": "train"
} |
math | [
{
"content": "How many pairs of two-digit positive integers have a difference of 50? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "8244",
"split": "train"
} |
math | [
{
"content": "The sequence $\\{a_i\\}_{i \\ge 1}$ is defined by $a_1 = 1$ and \\[ a_n = \\lfloor a_{n-1} + \\sqrt{a_{n-1}} \\rfloor \\] for all $n \\ge 2$ . Compute the eighth perfect square in the sequence.\n\n*Proposed by Lewis Chen* Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "8245",
"split": "train"
} |
math | [
{
"content": "In the two-dimensional rectangular coordinate system, given the vector $\\overrightarrow{a}=(-1,2)$, and points $A(8,0)$, $B(n,t)$, $C(k\\sin θ,t)(0≤θ≤\\frac {π}{2})$.\n(1) If $\\overrightarrow{AB} \\perp \\overrightarrow{a}$, and $|\\overrightarrow{AB}|= \\sqrt {5}|\\overrightarrow{OA}|(O$ is the... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "8246",
"split": "train"
} |
math | [
{
"content": "A teacher intends to give the children a problem of the following type. He will tell them that he has thought of a polynomial \\( P(x) \\) of degree 2017 with integer coefficients and a leading coefficient of 1. Then he will provide them with \\( k \\) integers \\( n_{1}, n_{2}, \\ldots, n_{k} \\)... | {
"ground_truth": "2017",
"style": "rule"
} | {
"index": "8247",
"split": "train"
} |
math | [
{
"content": "Represent the number 1000 as a sum of the maximum possible number of natural numbers, the sums of the digits of which are pairwise distinct. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "8248",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=x- \\frac {1}{x}+2a\\ln x$ $(a\\in\\mathbb{R})$.\n\n$(1)$ Discuss the monotonicity of $f(x)$;\n\n$(2)$ If $f(x)$ has two extreme values $x_{1}$ and $x_{2}$, where $x_{2}\\in[e,+\\infty)$, find the minimum value of $f(x_{1})-f(x_{2})$. Please output the final answer within ... | {
"ground_truth": "\\frac {4}{e}",
"style": "rule"
} | {
"index": "8249",
"split": "train"
} |
math | [
{
"content": "If a number is a multiple of 4 or contains the digit 4, we say this number is a \"4-inclusive number\", such as 20, 34. Arrange all \"4-inclusive numbers\" in the range \\[0, 100\\] in ascending order to form a sequence. What is the sum of all items in this sequence? Please output the final answer... | {
"ground_truth": "1883",
"style": "rule"
} | {
"index": "8250",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}=(2\\cos\\omega x,-2)$ and $\\overrightarrow{b}=(\\sqrt{3}\\sin\\omega x+\\cos\\omega x,1)$, where $\\omega\\ \\ \\gt 0$, and the function $f(x)=\\overrightarrow{a}\\cdot\\overrightarrow{b}+1$. The distance between two adjacent symmetric centers of the graph of $f... | {
"ground_truth": "\\frac{3-\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "8251",
"split": "train"
} |
math | [
{
"content": "At an old estate, the house is surrounded in a circle by tall trees: spruces, pines, and birches. There are a total of 96 trees. These trees have a strange property: from any coniferous tree, if you take two trees skipping one tree in between, one of them is coniferous and the other is deciduous; ... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "8252",
"split": "train"
} |
math | [
{
"content": "If $\\frac{\\sin\\theta + \\cos\\theta}{\\sin\\theta - \\cos\\theta} = 2$, calculate the value of $\\sin\\theta \\cdot \\cos\\theta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{10}",
"style": "rule"
} | {
"index": "8253",
"split": "train"
} |
math | [
{
"content": "When the polynomial $x^4 - 6x^3 + 16x^ 2 - 25x + 10$ is divided by $x^2 - 2x + k,$ the remainder is $x + a.$ Enter the ordered pair $(k,a).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(5,-5)",
"style": "rule"
} | {
"index": "8254",
"split": "train"
} |
math | [
{
"content": "Let an integer be such that when divided by 20, the remainder is 11. What is the sum of the remainders when the same integer is divided by 4 and by 5? Additionally, find the smallest such integer greater than 50. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "8255",
"split": "train"
} |
math | [
{
"content": "Given that \\(x\\) satisfies \\(\\log _{5x} (2x) = \\log _{625x} (8x)\\), find the value of \\(\\log _{2} x\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\ln 5}{2 \\ln 2 - 3 \\ln 5}",
"style": "rule"
} | {
"index": "8256",
"split": "train"
} |
math | [
{
"content": "Six balls, numbered 2, 3, 4, 5, 6, 7, are placed in a hat. Each ball is equally likely to be chosen. If one ball is chosen, what is the probability that the number on the selected ball is a prime number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "8257",
"split": "train"
} |
math | [
{
"content": "A street has 20 houses on each side, for a total of 40 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 4, 10, 16, etc., and on the north side they are 3, 9, 15, etc. A sign ... | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "8258",
"split": "train"
} |
math | [
{
"content": "Workshop A and Workshop B together have 360 workers. The number of workers in Workshop A is three times that of Workshop B. How many workers are there in each workshop? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "270",
"style": "rule"
} | {
"index": "8259",
"split": "train"
} |
math | [
{
"content": "Let the complex number \\( z = \\cos \\frac{2\\pi}{13} + i \\sin \\frac{2\\pi}{13} \\). Find the value of \\( \\left(z^{-12} + z^{-11} + z^{-10}\\right)\\left(z^{3} + 1\\right)\\left(z^{6} + 1\\right) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "8260",
"split": "train"
} |
math | [
{
"content": "Given that the terminal side of angle θ passes through point P(-x, -6) and $$cosθ=- \\frac {5}{13}$$, find the value of $$tan(θ+ \\frac {π}{4})$$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac {17}{7}",
"style": "rule"
} | {
"index": "8261",
"split": "train"
} |
math | [
{
"content": "If $\\lceil{\\sqrt{x}}\\rceil=12$, how many possible integer values of $x$ are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "8262",
"split": "train"
} |
math | [
{
"content": "Two positive integers that only have 1 as a common factor are called coprime numbers. For example, 2 and 7 are coprime, as are 3 and 4. In any permutation of 2, 3, 4, 5, 6, 7, where each pair of adjacent numbers are coprime, there are a total of \\_\\_\\_\\_\\_\\_\\_\\_ different permutations (ans... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "8263",
"split": "train"
} |
math | [
{
"content": "An ATM password at Fred's Bank is composed of four digits from $0$ to $9$, with repeated digits allowable. If no password may begin with the sequence $9,1,1,$ then how many passwords are possible? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9990",
"style": "rule"
} | {
"index": "8264",
"split": "train"
} |
math | [
{
"content": "In the right circular cone $P-ABC$, $PA \\perp$ plane $ABC$, $AC \\perp AB$, $PA=AB=2$, $AC=1$. Find the volume of the circumscribed sphere of the cone $P-ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{2}\\pi",
"style": "rule"
} | {
"index": "8265",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. The polar equation of line $l$ is $\\rho\\cos(\\theta+ \\frac{\\pi}{4})= \\frac{\\sqrt{2}}{2}$, and the parametric equation of curve... | {
"ground_truth": "\\sqrt{34}+2",
"style": "rule"
} | {
"index": "8266",
"split": "train"
} |
math | [
{
"content": "In quadrilateral $ABCD$, $\\angle{BAD}\\cong\\angle{ADC}$ and $\\angle{ABD}\\cong\\angle{BCD}$, $AB = 8$, $BD = 10$, and $BC = 6$. The length $CD$ may be written in the form $\\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. Please output the final answer with... | {
"ground_truth": "69",
"style": "rule"
} | {
"index": "8267",
"split": "train"
} |
math | [
{
"content": "When Dave walks to school, he averages $90$ steps per minute, and each of his steps is $75$ cm long. It takes him $16$ minutes to get to school. His brother, Jack, going to the same school by the same route, averages $100$ steps per minute, but his steps are only $60$ cm long. How long does it tak... | {
"ground_truth": "18 minutes",
"style": "rule"
} | {
"index": "8268",
"split": "train"
} |
math | [
{
"content": "Let \\( A_0 = (0,0) \\). Points \\( A_1, A_2, \\dots \\) lie on the \\( x \\)-axis, and distinct points \\( B_1, B_2, \\dots \\) lie on the graph of \\( y = x^2 \\). For every positive integer \\( n \\), \\( A_{n-1}B_nA_n \\) is an equilateral triangle. What is the least \\( n \\) for which the le... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "8269",
"split": "train"
} |
math | [
{
"content": "Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "x^4",
"style": "rule"
} | {
"index": "8270",
"split": "train"
} |
math | [
{
"content": "If a number $N, N \\ne 0$, diminished by four times its reciprocal, equals a given real constant $R$, then, for this given $R$, the sum of all such possible values of $N$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "R",
"style": "rule"
} | {
"index": "8271",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression \\[ \\frac{a+2}{a+1} \\cdot \\frac{b-1}{b-2} \\cdot \\frac{c + 8}{c+6} , \\] given that $c = b-10$, $b = a+2$, $a = 4$, and none of the denominators are zero. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "8272",
"split": "train"
} |
math | [
{
"content": "Sasha and Misha are playing a game: they take turns naming a number from 1 to 213 (Misha goes first, and the numbers must be different). Then each counts the number of different rectangles with integer sides whose perimeter equals the named number. The one with the greater number of rectangles win... | {
"ground_truth": "212",
"style": "rule"
} | {
"index": "8273",
"split": "train"
} |
math | [
{
"content": "The sequence $\\{a_n\\}$ satisfies $a_1=1$, $a_2=1$, $a_{n+2}=(1+\\sin^2 \\frac{n\\pi}{2})a_n+2\\cos^2 \\frac{n\\pi}{2}$. Find the sum of the first $20$ terms of this sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1123",
"style": "rule"
} | {
"index": "8274",
"split": "train"
} |
math | [
{
"content": "In which acute-angled triangle is the value of the product \\(\\operatorname{tg} \\alpha \\cdot \\operatorname{tg} \\beta \\cdot \\operatorname{tg} \\gamma\\) minimized? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{27}",
"style": "rule"
} | {
"index": "8275",
"split": "train"
} |
math | [
{
"content": "Consider a regular tetrahedron $ABCD$. Find $\\sin \\angle BAC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "8276",
"split": "train"
} |
math | [
{
"content": "For the nonzero numbers $a$, $b$, and $c$, define $$\n\\text{{J}}(a,b,c) = \\frac{a}{b} + \\frac{b}{c} + \\frac{c}{a}.\n$$Find $\\text{{J}}(2,12, 9)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "8277",
"split": "train"
} |
math | [
{
"content": "How many integer values of $n$ satisfy $-50 < n^3 < 50$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "8278",
"split": "train"
} |
math | [
{
"content": "Two candidates participated in an election with \\(p+q\\) voters. Candidate \\(A\\) received \\(p\\) votes and candidate \\(B\\) received \\(q\\) votes, with \\(p > q\\). During the vote counting, only one vote is registered at a time on a board. Let \\(r\\) be the probability that the number asso... | {
"ground_truth": "\\frac{1}{2019}",
"style": "rule"
} | {
"index": "8279",
"split": "train"
} |
math | [
{
"content": "Alec wishes to construct a string of 6 letters using the letters A, C, G, and N, such that: - The first three letters are pairwise distinct, and so are the last three letters; - The first, second, fourth, and fifth letters are pairwise distinct. In how many ways can he construct the string? Please... | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "8280",
"split": "train"
} |
math | [
{
"content": "A cube with side length $1$ is sliced by a plane that passes through two diagonally opposite vertices $A$ and $C$ and the midpoints $B$ and $D$ of two opposite edges not containing $A$ or $C$, as shown. What is the area of quadrilateral $ABCD$? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{\\sqrt{6}}{2}",
"style": "rule"
} | {
"index": "8281",
"split": "train"
} |
math | [
{
"content": "The mean (average) of 6, 9 and 18 is equal to the mean (average) of 12 and $y$. What is the value of $y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "8282",
"split": "train"
} |
math | [
{
"content": "Evaluate the determinant of the given matrix:\n\\[\n\\begin{vmatrix} \\sin \\theta \\sin \\phi & \\sin \\theta \\cos \\phi & \\cos \\theta \\\\ \\cos \\phi & -\\sin \\phi & 0 \\\\ -\\cos \\theta \\sin \\phi & -\\cos \\theta \\cos \\phi & \\sin \\theta \\end{vmatrix}\n\\] Please output the final an... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "8283",
"split": "train"
} |
math | [
{
"content": "In the equation \n$$\n\\frac{x^{2}+p}{x}=-\\frac{1}{4},\n$$\nwith roots \\(x_{1}\\) and \\(x_{2}\\), determine \\(p\\) such that:\n\na) \\(\\frac{x_{1}}{x_{2}}+\\frac{x_{2}}{x_{1}}=-\\frac{9}{4}\\),\n\nb) one root is 1 less than the square of the other root. Please output the final answer within \... | {
"ground_truth": "-\\frac{15}{8}",
"style": "rule"
} | {
"index": "8284",
"split": "train"
} |
math | [
{
"content": "In order to compute the area of a particular circle, Juan first measures the length of its diameter. The actual diameter is 20 cm, but Juan's measurement has an error of up to $20\\%$. What is the largest possible percent error, in percent, in Juan's computed area of the circle? Please output the ... | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "8285",
"split": "train"
} |
math | [
{
"content": "Given that the function $F(x) = f(x) + x^2$ is an odd function, and $f(2) = 1$, find $f(-2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-9",
"style": "rule"
} | {
"index": "8286",
"split": "train"
} |
math | [
{
"content": "Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by $10$ percent. Over the weekend, Lou advertises the sale: \"Ten percent off the listed price. Sale starts Monday.\" How much does a pair of shoes cost on Monday that co... | {
"ground_truth": "39.60",
"style": "rule"
} | {
"index": "8287",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate plane $(xOy)$, given vectors $\\overrightarrow{AB}=(6,1)$, $\\overrightarrow{BC}=(x,y)$, $\\overrightarrow{CD}=(-2,-3)$, and $\\overrightarrow{AD}$ is parallel to $\\overrightarrow{BC}$.\n(1) Find the relationship between $x$ and $y$;\n(2) If $\\overrightarrow{AC}$ is p... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "8288",
"split": "train"
} |
math | [
{
"content": "Compute $\\begin{pmatrix} \\sqrt{3} & -1 \\\\ 1 & \\sqrt{3} \\end{pmatrix}^6.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": " \\begin{pmatrix} -64 & 0 \\\\ 0 & -64 \\end{pmatrix} ",
"style": "rule"
} | {
"index": "8289",
"split": "train"
} |
math | [
{
"content": "Given sets $A=\\{1, 2, 3, 4\\}$, $B=\\{5, 6, 7\\}$, and $C=\\{8, 9\\}$. Now, take any two sets from these three sets, and then pick one element from each of the two selected sets to form a new set with two elements. The total number of distinct sets that can be formed is $\\_\\_\\_\\_\\_\\_\\_\\_$... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "8290",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of integers $n > 1$ for which $\\tfrac1n = 0.d_1d_2d_3d_4\\ldots$, an infinite decimal that has the property that $d_i = d_{i+12}$ for all positive integers $i$. Given that $9901$ is prime, how many positive integers are in $S$? (The $d_i$ are digits.) Please output the final an... | {
"ground_truth": "255",
"style": "rule"
} | {
"index": "8291",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $2a\\sin B= \\sqrt{3}b$ and $\\cos C = \\frac{5}{13}$: \n(1) Find the value of $\\sin A$; \n(2) Find the value of $\\cos B$. Please output the final answer within \\boxed{}.",... | {
"ground_truth": "\\frac{12\\sqrt{3} - 5}{26}",
"style": "rule"
} | {
"index": "8292",
"split": "train"
} |
math | [
{
"content": "Suppose $\\cos Q = 0.4$ in the diagram below. What is $QR$?\n\n\n\n[asy]\npair P,Q,R;\nP = (0,0);\nQ = (6,0);\nR = (0,6*tan(acos(0.4)));\ndraw(P--Q--R--P);\ndraw(rightanglemark(Q,P,R,18));\nlabel(\"$P$\",P,SW);\nlabel(\"$Q$\",Q,SE);\nlabel(\"$R$\",R,N);\nlabel(\"$12$\",Q/2,S);\n[/asy] Please outp... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "8293",
"split": "train"
} |
math | [
{
"content": "Express as a common fraction in simplest form: $$\n\\sqrt{6\\frac{1}{4}}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "8294",
"split": "train"
} |
math | [
{
"content": "A box contains 5 white balls and 5 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors, starting and ending with the same color? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{126}",
"style": "rule"
} | {
"index": "8295",
"split": "train"
} |
math | [
{
"content": "Use the bisection method to find an approximate zero of the function $f(x) = \\log x + x - 3$, given that approximate solutions (accurate to 0.1) are $\\log 2.5 \\approx 0.398$, $\\log 2.75 \\approx 0.439$, and $\\log 2.5625 \\approx 0.409$. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "2.6",
"style": "rule"
} | {
"index": "8296",
"split": "train"
} |
math | [
{
"content": "In a bag of marbles, $\\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{7}",
"style": "rule"
} | {
"index": "8297",
"split": "train"
} |
math | [
{
"content": "Let vectors $\\overrightarrow{a_{1}}=(1,5)$, $\\overrightarrow{a_{2}}=(4,-1)$, $\\overrightarrow{a_{3}}=(2,1)$, and let $\\lambda_{1}, \\lambda_{2}, \\lambda_{3}$ be non-negative real numbers such that $\\lambda_{1}+\\frac{\\lambda_{2}}{2}+\\frac{\\lambda_{3}}{3}=1$. Find the minimum value of $\\l... | {
"ground_truth": "3\\sqrt{2}",
"style": "rule"
} | {
"index": "8298",
"split": "train"
} |
math | [
{
"content": "Bricklayer Brenda would take $9$ hours to build a chimney alone, and bricklayer Brandon would take $10$ hours to build it alone. When they work together they talk a lot, and their combined output is decreased by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. How many ... | {
"ground_truth": "900",
"style": "rule"
} | {
"index": "8299",
"split": "train"
} |
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