data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In the year 2009, there is a property that rearranging the digits of the number 2009 cannot yield a smaller four-digit number (numbers do not start with zero). In what subsequent year does this property first repeat again? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2022",
"style": "rule"
} | {
"index": "2100",
"split": "train"
} |
math | [
{
"content": "Calculate the value of $(2345 + 3452 + 4523 + 5234) \\times 2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31108",
"style": "rule"
} | {
"index": "2101",
"split": "train"
} |
math | [
{
"content": "Find the 6-digit repetend in the decimal representation of $\\frac{7}{29}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "241379",
"style": "rule"
} | {
"index": "2102",
"split": "train"
} |
math | [
{
"content": "Joey and his five brothers are ages $3$, $5$, $7$, $9$, $11$, and $13$. One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the $5$-year-old stayed home. How old is Joey? Please output the final answer with... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "2103",
"split": "train"
} |
math | [
{
"content": "The average age of five children is 6 years old. Each child is a different age and there is a difference of two years in the ages of any two consecutive children. In years, how old is the oldest child? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "2104",
"split": "train"
} |
math | [
{
"content": "An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of 1 unit either up or to the right. A lattice point $(x, y)$ with $0 \\leq x, y \\leq 5$ is chosen uniformly at random. Compute the expected number of up-right paths from $(0,0)$ to $(5,5)$ not pass... | {
"ground_truth": "175",
"style": "rule"
} | {
"index": "2105",
"split": "train"
} |
math | [
{
"content": "Find the area of quadrilateral \\(ABCD\\) if \\(AB = BC = 3\\sqrt{3}\\), \\(AD = DC = \\sqrt{13}\\), and vertex \\(D\\) lies on a circle of radius 2 inscribed in the angle \\(ABC\\), where \\(\\angle ABC = 60^\\circ\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{3}",
"style": "rule"
} | {
"index": "2106",
"split": "train"
} |
math | [
{
"content": "A certain model of hybrid car travels from point $A$ to point $B$ with a fuel cost of $76$ yuan, and with an electricity cost of $26 yuan. It is known that for every kilometer traveled, the fuel cost is $0.5$ yuan more than the electricity cost. \n$(1)$ Find the cost of traveling one kilometer us... | {
"ground_truth": "74",
"style": "rule"
} | {
"index": "2107",
"split": "train"
} |
math | [
{
"content": "What is the area, in square units, of triangle $ABC$? [asy]\nunitsize(0.15inch);\npath X = (-6.5, 0)--(5.5, 0);\npath Y = (0, -3.5)--(0, 7.5);\ndraw(X); draw(Y);\n\nfor(int n=-6; n <= 5; ++n)\nif( n != 0 )\ndraw( (n,0.25)--(n,-0.25) );\nfor(int n=-3; n <= 7; ++n)\nif( n != 0 )\ndraw( (0.25,n)--(-0... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "2108",
"split": "train"
} |
math | [
{
"content": "The matrix $\\mathbf{A} = \\begin{pmatrix} 2 & 3 \\\\ 5 & d \\end{pmatrix}$ satisfies\n\\[\\mathbf{A}^{-1} = k \\mathbf{A}\\]for some constant $k.$ Enter the ordered pair $(d,k).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left( -2, \\frac{1}{19} \\right)",
"style": "rule"
} | {
"index": "2109",
"split": "train"
} |
math | [
{
"content": "Olga Ivanovna, the homeroom teacher of class 5B, is organizing a \"Mathematical Ballet\". She wants to arrange the boys and girls so that exactly 2 boys are at a distance of 5 meters from each girl. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys a... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "2110",
"split": "train"
} |
math | [
{
"content": "To make pizza dough, Luca mixes 50 mL of milk for every 250 mL of flour. How many mL of milk does he mix with 750 mL of flour? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "2111",
"split": "train"
} |
math | [
{
"content": "Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-10",
"style": "rule"
} | {
"index": "2112",
"split": "train"
} |
math | [
{
"content": "The jury, when preparing versions of the district math olympiad problems for grades $7, 8, 9, 10, 11$, aims to ensure that each version for each grade contains exactly 7 problems, of which exactly 4 do not appear in any other version. What is the maximum number of problems that can be included in ... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "2113",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 1.$ Find the minimum value of\n\\[(x + 2y)(y + 2z)(xz + 1).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "2114",
"split": "train"
} |
math | [
{
"content": "The positive integers $m$ and $n$ satisfy $8m + 9n = mn + 6$. Find the maximum value of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "2115",
"split": "train"
} |
math | [
{
"content": "Tim's quiz scores were 85, 87, 92, 94, 78, and 96. Calculate his mean score and find the range of his scores. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "2116",
"split": "train"
} |
math | [
{
"content": "A class leader is planning to invite graduates from the class of 2016 to give speeches. Out of 8 people, labeled A, B, ..., H, the leader wants to select 4 to speak. The conditions are: (1) at least one of A and B must participate; (2) if both A and B participate, there must be exactly one person ... | {
"ground_truth": "1080",
"style": "rule"
} | {
"index": "2117",
"split": "train"
} |
math | [
{
"content": "A circle passes through the vertices $K$ and $P$ of triangle $KPM$ and intersects its sides $KM$ and $PM$ at points $F$ and $B$, respectively. Given that $K F : F M = 3 : 1$ and $P B : B M = 6 : 5$, find $K P$ given that $B F = \\sqrt{15}$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "2 \\sqrt{33}",
"style": "rule"
} | {
"index": "2118",
"split": "train"
} |
math | [
{
"content": "When 1524 shi of rice is mixed with an unknown amount of wheat, and in a sample of 254 grains, 28 are wheat grains, calculate the estimated amount of wheat mixed with this batch of rice. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "168",
"style": "rule"
} | {
"index": "2119",
"split": "train"
} |
math | [
{
"content": "Medians $\\overline{DP}$ and $\\overline{EQ}$ of $\\triangle DEF$ intersect at an angle of $60^\\circ$. If $DP = 21$ and $EQ = 27$, determine the length of side $DE$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{67}",
"style": "rule"
} | {
"index": "2120",
"split": "train"
} |
math | [
{
"content": "Given that \\(\\triangle ABC\\) has sides \\(a\\), \\(b\\), and \\(c\\) corresponding to angles \\(A\\), \\(B\\), and \\(C\\) respectively, and knowing that \\(a + b + c = 16\\), find the value of \\(b^2 \\cos^2 \\frac{C}{2} + c^2 \\cos^2 \\frac{B}{2} + 2bc \\cos \\frac{B}{2} \\cos \\frac{C}{2} \\... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "2121",
"split": "train"
} |
math | [
{
"content": "In the hexagonal pyramid $(P-ABCDEF)$, the base is a regular hexagon with side length $\\sqrt{2}$, $PA=2$ and is perpendicular to the base. Find the volume of the circumscribed sphere of the hexagonal pyramid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{3}\\pi",
"style": "rule"
} | {
"index": "2122",
"split": "train"
} |
math | [
{
"content": "Each of $a, b$ and $c$ is equal to a number from the list $3^{1}, 3^{2}, 3^{3}, 3^{4}, 3^{5}, 3^{6}, 3^{7}, 3^{8}$. There are $N$ triples $(a, b, c)$ with $a \\leq b \\leq c$ for which each of $\\frac{ab}{c}, \\frac{ac}{b}$ and $\\frac{bc}{a}$ is equal to an integer. What is the value of $N$? Plea... | {
"ground_truth": "86",
"style": "rule"
} | {
"index": "2123",
"split": "train"
} |
math | [
{
"content": "In a classroom, 34 students are seated in 5 rows of 7 chairs. The place at the center of the room is unoccupied. A teacher decides to reassign the seats such that each student will occupy a chair adjacent to his/her present one (i.e. move one desk forward, back, left or right). In how many ways ca... | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "2124",
"split": "train"
} |
math | [
{
"content": "The maximum value of the function $f(x) = \\frac{\\frac{1}{6} \\cdot (-1)^{1+ C_{2x}^{x}} \\cdot A_{x+2}^{5}}{1+ C_{3}^{2} + C_{4}^{2} + \\ldots + C_{x-1}^{2}}$ ($x \\in \\mathbb{N}$) is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-20",
"style": "rule"
} | {
"index": "2125",
"split": "train"
} |
math | [
{
"content": "Express the given data \"$20$ nanoseconds\" in scientific notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\times 10^{-8}",
"style": "rule"
} | {
"index": "2126",
"split": "train"
} |
math | [
{
"content": "Simplify the expression $\\dfrac{45}{28} \\cdot \\dfrac{49}{75} \\cdot \\dfrac{100}{63}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} | {
"index": "2127",
"split": "train"
} |
math | [
{
"content": "Given that $x \\in (1,5)$, find the minimum value of the function $y= \\frac{2}{x-1}+ \\frac{1}{5-x}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3+2 \\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "2128",
"split": "train"
} |
math | [
{
"content": "Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1440",
"style": "rule"
} | {
"index": "2129",
"split": "train"
} |
math | [
{
"content": "A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\\heartsuit$ and $\\diamondsuit$, called `hearts' and `diamonds') are red, the other two ($\\spadesuit$ and $\\clubsuit$, called `spades' and `clubs') are black. The cards in the deck are p... | {
"ground_truth": "\\frac{25}{102}",
"style": "rule"
} | {
"index": "2130",
"split": "train"
} |
math | [
{
"content": "What non-zero, real value of $x$ satisfies $(5x)^4= (10x)^3$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{5}",
"style": "rule"
} | {
"index": "2131",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ $c$ be the roots of the cubic polynomial $x^3 - x - 1 = 0.$ Find\n\\[a(b - c)^2 + b(c - a)^2 + c(a - b)^2.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-9",
"style": "rule"
} | {
"index": "2132",
"split": "train"
} |
math | [
{
"content": "Simplify $(1 + \\tan 20^\\circ)(1 + \\tan 25^\\circ).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "2133",
"split": "train"
} |
math | [
{
"content": "A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "2134",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a=2b$. Also, $\\sin A$, $\\sin C$, $\\sin B$ form an arithmetic sequence.\n\n$(I)$ Find the value of $\\cos (B+C)$;\n\n$(II)$ If the area of $\\triangle ABC$ is $\\frac{8\\sqrt{15}}{3}$, find th... | {
"ground_truth": "4 \\sqrt {2}",
"style": "rule"
} | {
"index": "2135",
"split": "train"
} |
math | [
{
"content": "$2(81+83+85+87+89+91+93+95+97+99)= $ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1800",
"style": "rule"
} | {
"index": "2136",
"split": "train"
} |
math | [
{
"content": "Given that $α$ is an acute angle and $\\sin α= \\frac {3}{5}$, find the value of $\\cos α$ and $\\cos (α+ \\frac {π}{6})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {4\\sqrt {3}-3}{10}",
"style": "rule"
} | {
"index": "2137",
"split": "train"
} |
math | [
{
"content": "A trauma hospital uses a rectangular piece of white cloth that is 60m long and 0.8m wide to make triangular bandages with both base and height of 0.4m. How many bandages can be made in total? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "2138",
"split": "train"
} |
math | [
{
"content": "Petya and Vasya are playing the following game. Petya chooses a non-negative random value $\\xi$ with expectation $\\mathbb{E} [\\xi ] = 1$ , after which Vasya chooses his own value $\\eta$ with expectation $\\mathbb{E} [\\eta ] = 1$ without reference to the value of $\\xi$ . For which max... | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "2139",
"split": "train"
} |
math | [
{
"content": "Two trains are moving towards each other on parallel tracks - one with a speed of 60 km/h and the other with a speed of 80 km/h. A passenger sitting in the second train noticed that the first train passed by him in 6 seconds. What is the length of the first train? Please output the final answer wi... | {
"ground_truth": "233.33",
"style": "rule"
} | {
"index": "2140",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = \\left\\lceil\\dfrac{1}{x+2}\\right\\rceil$ for $x > -2$, and $f(x) = \\left\\lfloor\\dfrac{1}{x+2}\\right\\rfloor$ for $x < -2$. ($f(x)$ is not defined at $x = -2$.) Which integer is not in the range of $f(x)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "2141",
"split": "train"
} |
math | [
{
"content": "Given that point $A(1,1)$ is a point on the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and $F\\_1$, $F\\_2$ are the two foci of the ellipse such that $|AF\\_1|+|AF\\_2|=4$.\n(I) Find the standard equation of the ellipse;\n(II) Find the equation of the tangent line to the e... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "2142",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin \\alpha $ and $\\cos \\alpha $ are the two roots of the quadratic equation $2x^{2}-x-m=0$ with respect to $x$, find $\\sin \\alpha +\\cos \\alpha =$______ and $m=\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "2143",
"split": "train"
} |
math | [
{
"content": "Niffy's favorite number is a positive integer, and Stebbysaurus is trying to guess what it is. Niffy tells her that when expressed in decimal without any leading zeros, her favorite number satisfies the following: - Adding 1 to the number results in an integer divisible by 210 . - The sum of the d... | {
"ground_truth": "1010309",
"style": "rule"
} | {
"index": "2144",
"split": "train"
} |
math | [
{
"content": "If $f(a)=a-2$ and $F(a,b)=b^2+a$, then $F(3,f(4))$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "2145",
"split": "train"
} |
math | [
{
"content": "An object moves $8$ cm in a straight line from $A$ to $B$, turns at an angle $\\alpha$, measured in radians and chosen at random from the interval $(0,\\pi)$, and moves $5$ cm in a straight line to $C$. What is the probability that $AC < 7$? Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "2146",
"split": "train"
} |
math | [
{
"content": "The sum of the digits of the integer equal to \\( 777777777777777^2 - 222222222222223^2 \\) can be found by evaluating the expression. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "74",
"style": "rule"
} | {
"index": "2147",
"split": "train"
} |
math | [
{
"content": "What is $2^{-1} + 2^{-2} + 2^{-3} + 2^{-4} + 2^{-5} + 2^{-6} \\pmod{13}$?\n\nExpress your answer as an integer from $0$ to $12$, inclusive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "2148",
"split": "train"
} |
math | [
{
"content": "A rectangle has a perimeter of 80 inches and an area greater than 240 square inches. How many non-congruent rectangles meet these criteria? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "2149",
"split": "train"
} |
math | [
{
"content": "$\\triangle ABC$ has a right angle at $C$ and $\\angle A = 20^\\circ$. If $BD$ ($D$ in $\\overline{AC}$) is the bisector of $\\angle ABC$, then $\\angle BDC =$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "55^{\\circ}",
"style": "rule"
} | {
"index": "2150",
"split": "train"
} |
math | [
{
"content": "Given $f(x)=9^{x}-2×3^{x}+4$, where $x\\in\\[-1,2\\]$:\n1. Let $t=3^{x}$, with $x\\in\\[-1,2\\}$, find the maximum and minimum values of $t$.\n2. Find the maximum and minimum values of $f(x)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "2151",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $1-\\frac{a-b}{a+2b}÷\\frac{a^2-b^2}{a^2+4ab+4b^2}$. Given that $a=2\\sin 60^{\\circ}-3\\tan 45^{\\circ}$ and $b=3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\sqrt{3}",
"style": "rule"
} | {
"index": "2152",
"split": "train"
} |
math | [
{
"content": "Given real numbers $x$, $y$, $z$ satisfying $2x-y-2z-6=0$, and $x^2+y^2+z^2\\leq4$, calculate the value of $2x+y+z$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "2153",
"split": "train"
} |
math | [
{
"content": "The probability that the random variable $X$ follows a normal distribution $N\\left( 3,{{\\sigma }^{2}} \\right)$ and $P\\left( X\\leqslant 4 \\right)=0.84$ can be expressed in terms of the standard normal distribution $Z$ as $P(Z\\leqslant z)=0.84$, where $z$ is the z-score corresponding to the u... | {
"ground_truth": "0.68",
"style": "rule"
} | {
"index": "2154",
"split": "train"
} |
math | [
{
"content": "In △ABC, B = $$\\frac{\\pi}{3}$$, AB = 8, BC = 5, find the area of the circumcircle of △ABC. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{49\\pi}{3}",
"style": "rule"
} | {
"index": "2155",
"split": "train"
} |
math | [
{
"content": "Determine the heaviest weight that can be obtained using a combination of 2 lb and 4 lb, and 12 lb weights with a maximum of two weights used at a time. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "2156",
"split": "train"
} |
math | [
{
"content": "A parallelogram-shaped paper WXYZ with an area of 7.17 square centimeters is placed on another parallelogram-shaped paper EFGH, as shown in the diagram. The intersection points A, C, B, and D are formed, and AB // EF and CD // WX. What is the area of the paper EFGH in square centimeters? Explain t... | {
"ground_truth": "7.17",
"style": "rule"
} | {
"index": "2157",
"split": "train"
} |
math | [
{
"content": "Let \\(A B C D\\) be a square of side length 13. Let \\(E\\) and \\(F\\) be points on rays \\(A B\\) and \\(A D\\), respectively, so that the area of square \\(A B C D\\) equals the area of triangle \\(A E F\\). If \\(E F\\) intersects \\(B C\\) at \\(X\\) and \\(B X = 6\\), determine \\(D F\\). P... | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "2158",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a\\sin 2B= \\sqrt{3}b\\sin A$.\n\n1. Find $B$;\n2. If $\\cos A= \\dfrac{1}{3}$, find the value of $\\sin C$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "\\dfrac{2\\sqrt{6}+1}{6}",
"style": "rule"
} | {
"index": "2159",
"split": "train"
} |
math | [
{
"content": "If two distinct numbers are selected at random from the first seven prime numbers, what is the probability that their sum is an even number? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{7}",
"style": "rule"
} | {
"index": "2160",
"split": "train"
} |
math | [
{
"content": "If a number eight times as large as $x$ is increased by two, then one fourth of the result equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2x + \\frac{1}{2}",
"style": "rule"
} | {
"index": "2161",
"split": "train"
} |
math | [
{
"content": "Given the quadratic function $f(x)=x^{2}+mx+n$.\n(1) If $f(x)$ is an even function with a minimum value of $1$, find the analytic expression of $f(x)$;\n(2) Under the condition of (1), for the function $g(x)= \\frac {6x}{f(x)}$, solve the inequality $g(2^{x}) > 2^{x}$ with respect to $x$;\n(3) For... | {
"ground_truth": "\\frac {1}{2}",
"style": "rule"
} | {
"index": "2162",
"split": "train"
} |
math | [
{
"content": "Let $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set $$ {10a+b:1\\leq a\\leq 5, 1\\leq b\\leq 5} $$ where $a,b$ are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder wh... | {
"ground_truth": "47",
"style": "rule"
} | {
"index": "2163",
"split": "train"
} |
math | [
{
"content": "What is the sum of the eight terms in the arithmetic sequence $-2, 3, \\dots, 33$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "124",
"style": "rule"
} | {
"index": "2164",
"split": "train"
} |
math | [
{
"content": "A positive integer $a$ is input into a machine. If $a$ is odd, the output is $a+3$. If $a$ is even, the output is $a+5$. This process can be repeated using each successive output as the next input. If the input is $a=15$ and the machine is used 51 times, what is the final output? Please output the... | {
"ground_truth": "218",
"style": "rule"
} | {
"index": "2165",
"split": "train"
} |
math | [
{
"content": "Given two spherical balls of different sizes placed in two corners of a rectangular room, where each ball touches two walls and the floor, and there is a point on each ball such that the distance from the two walls it touches to that point is 5 inches and the distance from the floor to that point ... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "2166",
"split": "train"
} |
math | [
{
"content": "Consider the polynomials \\[f(x)=1-12x+3x^2-4x^3+5x^4\\] and \\[g(x)=3-2x-6x^3+9x^4.\\] Find $c$ such that the polynomial $f(x)+cg(x)$ has degree 3. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{5}{9}",
"style": "rule"
} | {
"index": "2167",
"split": "train"
} |
math | [
{
"content": "Given $ \\frac {\\pi}{2} < \\alpha < \\pi$ and $0 < \\beta < \\frac {\\pi}{2}$, with $\\tan \\alpha= -\\frac {3}{4}$ and $\\cos (\\beta-\\alpha)= \\frac {5}{13}$, find the value of $\\sin \\beta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {63}{65}",
"style": "rule"
} | {
"index": "2168",
"split": "train"
} |
math | [
{
"content": "One night, 21 people exchanged phone calls $n$ times. It is known that among these people, there are $m$ people $a_{1}, a_{2}, \\cdots, a_{m}$ such that $a_{i}$ called $a_{i+1}$ (for $i=1,2, \\cdots, m$ and $a_{m+1}=a_{1}$), and $m$ is an odd number. If no three people among these 21 people have a... | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "2169",
"split": "train"
} |
math | [
{
"content": "What is the nearest integer to $(3+\\sqrt2)^6$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7414",
"style": "rule"
} | {
"index": "2170",
"split": "train"
} |
math | [
{
"content": "There is a set of data: $a_{1}=\\frac{3}{1×2×3}$, $a_{2}=\\frac{5}{2×3×4}$, $a_{3}=\\frac{7}{3×4×5}$, $\\ldots $, $a_{n}=\\frac{2n+1}{n(n+1)(n+2)}$. Let $S_{n}=a_{1}+a_{2}+a_{3}+\\ldots +a_{n}$. Find the value of $S_{12}$. To solve this problem, Xiao Ming first simplified $a_{n}$ to $a_{n}=\\frac{... | {
"ground_truth": "\\frac{201}{182}",
"style": "rule"
} | {
"index": "2171",
"split": "train"
} |
math | [
{
"content": "A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{20}",
"style": "rule"
} | {
"index": "2172",
"split": "train"
} |
math | [
{
"content": "An organization has a structure where there is one president, two vice-presidents (VP1 and VP2), and each vice-president supervises two managers. If the organization currently has 12 members, in how many different ways can the leadership (president, vice-presidents, and managers) be chosen? Please... | {
"ground_truth": "554400",
"style": "rule"
} | {
"index": "2173",
"split": "train"
} |
math | [
{
"content": "The students' written work has a binary grading system, i.e., a work will either be accepted if it is done well or not accepted if done poorly. Initially, the works are checked by a neural network which makes an error in 10% of the cases. All works identified as not accepted by the neural network ... | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "2174",
"split": "train"
} |
math | [
{
"content": "A bag contains 10 red marbles and 6 blue marbles. Three marbles are selected at random and without replacement. What is the probability that one marble is red and two are blue? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{15}{56}",
"style": "rule"
} | {
"index": "2175",
"split": "train"
} |
math | [
{
"content": "Given the ellipse Q: $$\\frac{x^{2}}{a^{2}} + y^{2} = 1 \\quad (a > 1),$$ where $F_{1}$ and $F_{2}$ are its left and right foci, respectively. A circle with the line segment $F_{1}F_{2}$ as its diameter intersects the ellipse Q at exactly two points. \n(1) Find the equation of ellipse Q; \n(2) S... | {
"ground_truth": "\\frac{3\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "2176",
"split": "train"
} |
math | [
{
"content": "The base-10 numbers 217 and 45 are multiplied. The product is then written in base-6. What is the units digit of the base-6 representation? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "2177",
"split": "train"
} |
math | [
{
"content": "Elena earns $\\$ 13.25$ per hour working at a store. How much does Elena earn in 4 hours? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\$53.00",
"style": "rule"
} | {
"index": "2178",
"split": "train"
} |
math | [
{
"content": "For positive integers \\( n \\), let \\( g(n) \\) return the smallest positive integer \\( k \\) such that \\( \\frac{1}{k} \\) has exactly \\( n \\) digits after the decimal point in base 6 notation. Determine the number of positive integer divisors of \\( g(2023) \\). Please output the final ans... | {
"ground_truth": "4096576",
"style": "rule"
} | {
"index": "2179",
"split": "train"
} |
math | [
{
"content": "Given the expansion of $(\\sqrt{x} + \\frac{2}{x^2})^n$, the ratio of the coefficient of the fifth term to the coefficient of the third term is 56:3. \n(Ⅰ) Find the constant term in the expansion; \n(Ⅱ) When $x=4$, find the term with the maximum binomial coefficient in the expansion. Please outp... | {
"ground_truth": "\\frac{63}{256}",
"style": "rule"
} | {
"index": "2180",
"split": "train"
} |
math | [
{
"content": "A train departs from point $A$ and travels towards point $B$ at a uniform speed. 11 minutes later, another train departs from point $B$ traveling towards point $A$ at a constant speed on the same route. After their meeting point, it takes the trains 20 minutes and 45 minutes to reach $B$ and $A$, ... | {
"ground_truth": "9/5",
"style": "rule"
} | {
"index": "2181",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $AB=13$, $BC=15$ and $CA=17$. Point $D$ is on $\\overline{AB}$, $E$ is on $\\overline{BC}$, and $F$ is on $\\overline{CA}$. Let $AD=p\\cdot AB$, $BE=q\\cdot BC$, and $CF=r\\cdot CA$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=2/3$ and $p^2+q^2+r^2=2/5$. The ratio of... | {
"ground_truth": "61",
"style": "rule"
} | {
"index": "2182",
"split": "train"
} |
math | [
{
"content": "Rationalize the denominator of $\\displaystyle \\frac{1}{\\sqrt[3]{3} - \\sqrt[3]{2}}$. With your answer in the form $\\displaystyle \\frac{\\sqrt[3]{A} + \\sqrt[3]{B} + \\sqrt[3]{C}}{D}$, and the fraction in lowest terms, what is $A + B + C + D$? Please output the final answer within \\boxed{}.",... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "2183",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC,$ $\\cot A \\cot C = \\frac{1}{2}$ and $\\cot B \\cot C = \\frac{1}{18}.$ Find $\\tan C.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "2184",
"split": "train"
} |
math | [
{
"content": "An infinite sequence $ \\,x_{0},x_{1},x_{2},\\ldots \\,$ of real numbers is said to be [b]bounded[/b] if there is a constant $ \\,C\\,$ such that $ \\, \\vert x_{i} \\vert \\leq C\\,$ for every $ \\,i\\geq 0$. Given any real number $ \\,a > 1,\\,$ construct a bounded infinite sequence $ x_{0},x_{1... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "2185",
"split": "train"
} |
math | [
{
"content": "Points \\( C_1 \\), \\( A_1 \\), and \\( B_1 \\) are taken on the sides \\( AB \\), \\( BC \\), and \\( AC \\) of triangle \\( ABC \\) respectively, such that\n\n\\[\n\\frac{AC_1}{C_1B} = \\frac{BA_1}{A_1C} = \\frac{CB_1}{B_1A} = 2.\n\\]\n\nFind the area of triangle \\( A_1B_1C_1 \\) if the area o... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "2186",
"split": "train"
} |
math | [
{
"content": "The sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$n(n+2)$",
"style": "rule"
} | {
"index": "2187",
"split": "train"
} |
math | [
{
"content": "The number of terms in the expansion of $[(a+3b)^{2}(a-3b)^{2}]^{2}$ when simplified is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "2188",
"split": "train"
} |
math | [
{
"content": "Three Graces each had the same number of fruits and met 9 Muses. Each Grace gave an equal number of fruits to each Muse. After that, each Muse and each Grace had the same number of fruits. How many fruits did each Grace have before meeting the Muses? Please output the final answer within \\boxed{}... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "2189",
"split": "train"
} |
math | [
{
"content": "Among A, B, C, and D comparing their heights, the sum of the heights of two of them is equal to the sum of the heights of the other two. The average height of A and B is 4 cm more than the average height of A and C. D is 10 cm taller than A. The sum of the heights of B and C is 288 cm. What is the... | {
"ground_truth": "139",
"style": "rule"
} | {
"index": "2190",
"split": "train"
} |
math | [
{
"content": "Given an ellipse $E$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\\frac{\\sqrt{3}}{2}$ and a minor axis length of $2$.\n1. Find the equation of the ellipse $E$;\n2. A line $l$ is tangent to a circle $C$: $x^{2}+y^{2}=r^{2}(0 < r < b)$ at any point and inters... | {
"ground_truth": "\\frac{2\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "2191",
"split": "train"
} |
math | [
{
"content": "Each of \\( A \\), \\( B \\), \\( C \\), and \\( D \\) is a positive two-digit integer. These integers satisfy each of the equations\n\n\\[\n\\begin{aligned}\nB &= 3C \\\\\nD &= 2B - C \\\\\nA &= B + D\n\\end{aligned}\n\\]\n\nWhat is the largest possible value of \\( A + B + C + D \\)? Please outp... | {
"ground_truth": "204",
"style": "rule"
} | {
"index": "2192",
"split": "train"
} |
math | [
{
"content": "What is the remainder when $2^{202} + 202$ is divided by $2^{101} + 2^{51} + 1$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "201",
"style": "rule"
} | {
"index": "2193",
"split": "train"
} |
math | [
{
"content": "A right rectangular prism $Q$ has integral side lengths $a, b, c$ with $a \\le b \\le c$. A plane parallel to one of the faces of $Q$ cuts $Q$ into two prisms, one of which is similar to $Q$, with both having nonzero volumes. The middle side length $b = 3969$. Determine the number of ordered tripl... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "2194",
"split": "train"
} |
math | [
{
"content": "Determine the least possible value of $f(1998),$ where $f:\\Bbb{N}\\to \\Bbb{N}$ is a function such that for all $m,n\\in {\\Bbb N}$, \n\n\\[f\\left( n^{2}f(m)\\right) =m\\left( f(n)\\right) ^{2}. \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "2195",
"split": "train"
} |
math | [
{
"content": "Let $f(t)=\\frac{t}{1-t}$, $t \\not= 1$. If $y=f(x)$, then $x$ can be expressed as Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-f(-y)",
"style": "rule"
} | {
"index": "2196",
"split": "train"
} |
math | [
{
"content": "Define: \\( a \\oplus b = a \\times b \\), \\( c \\bigcirc d = d \\times d \\times d \\times \\cdots \\times d \\) (d multiplied c times). Find \\( (5 \\oplus 8) \\oplus (3 \\bigcirc 7) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13720",
"style": "rule"
} | {
"index": "2197",
"split": "train"
} |
math | [
{
"content": "The pentagon \\(PQRST\\) is divided into four triangles with equal perimeters. The triangle \\(PQR\\) is equilateral. \\(PTU\\), \\(SUT\\), and \\(RSU\\) are congruent isosceles triangles. What is the ratio of the perimeter of the pentagon \\(PQRST\\) to the perimeter of the triangle \\(PQR\\)? Pl... | {
"ground_truth": "5:3",
"style": "rule"
} | {
"index": "2198",
"split": "train"
} |
math | [
{
"content": "One interior angle of a convex polygon is 160 degrees. The rest of the interior angles of the polygon are each 112 degrees. How many sides does the polygon have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "2199",
"split": "train"
} |
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