data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "The sequence 1,3,1,3,3,1,3,3,3,1,3,3,3,3,1,3,... follows a certain rule. What is the sum of the first 44 terms in this sequence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "116",
"style": "rule"
} | {
"index": "34300",
"split": "train"
} |
math | [
{
"content": "Let $A, M$, and $C$ be digits with $(100A+10M+C)(A+M+C) = 2005$. What is $A$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "34301",
"split": "train"
} |
math | [
{
"content": "Given $$\\alpha, \\beta \\in (0, \\frac{\\pi}{2})$$, and $$\\alpha + \\beta \\neq \\frac{\\pi}{2}, \\sin\\beta = \\sin\\alpha\\cos(\\alpha + \\beta)$$.\n(1) Express $\\tan\\beta$ in terms of $\\tan\\alpha$;\n(2) Find the maximum value of $\\tan\\beta$. Please output the final answer within \\boxed... | {
"ground_truth": "\\frac{\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "34302",
"split": "train"
} |
math | [
{
"content": "Let $P$ be the plane passing through the origin with normal vector $\\begin{pmatrix} 1 \\\\ -2 \\\\ 1 \\end{pmatrix}.$ Find the matrix $\\mathbf{P}$ such that for any vector $\\mathbf{v},$ $\\mathbf{P} \\mathbf{v}$ is the projection of $\\mathbf{v}$ onto plane $P.$ Please output the final answer ... | {
"ground_truth": "\\begin{pmatrix} \\frac{5}{6} & \\frac{1}{3} & -\\frac{1}{6} \\\\ \\frac{1}{3} & \\frac{1}{3} & \\frac{1}{3} \\\\ -\\frac{1}{6} & \\frac{1}{3} & \\frac{5}{6} \\end{pmatrix}",
"style": "rule"
} | {
"index": "34303",
"split": "train"
} |
math | [
{
"content": "The area of the square is $s^2$ and the area of the rectangle is $3s \\times \\frac{9s}{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7.41\\%",
"style": "rule"
} | {
"index": "34304",
"split": "train"
} |
math | [
{
"content": "Jefferson Middle School has the same number of boys and girls. $\\frac{3}{4}$ of the girls and $\\frac{2}{3}$ of the boys went on a field trip. What fraction of the students on the field trip were girls? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{17}",
"style": "rule"
} | {
"index": "34305",
"split": "train"
} |
math | [
{
"content": "Points $A$ , $B$ , $C$ , $D$ , and $E$ are on the same plane such that $A,E,C$ lie on a line in that order, $B,E,D$ lie on a line in that order, $AE = 1$ , $BE = 4$ , $CE = 3$ , $DE = 2$ , and $\\angle AEB = 60^\\circ$ . Let $AB$ and $CD$ intersect at $P$ . The square of the ar... | {
"ground_truth": "967",
"style": "rule"
} | {
"index": "34306",
"split": "train"
} |
math | [
{
"content": "What is the value of $(625^{\\log_5 2015})^{\\frac{1}{4}}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2015",
"style": "rule"
} | {
"index": "34307",
"split": "train"
} |
math | [
{
"content": "There are two alloys of copper and zinc. In the first alloy, there is twice as much copper as zinc, and in the second alloy, there is five times less copper than zinc. In what ratio should these alloys be combined to obtain a new alloy in which zinc is twice as much as copper? Please output the fi... | {
"ground_truth": "1 : 2",
"style": "rule"
} | {
"index": "34308",
"split": "train"
} |
math | [
{
"content": "Determine the form of $n$ such that $2^n + 2$ is divisible by $n$ where $n$ is less than 100. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "n=6, 66, 946",
"style": "rule"
} | {
"index": "34309",
"split": "train"
} |
math | [
{
"content": "The route not passing through the Zoo is 11 times shorter. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "34310",
"split": "train"
} |
math | [
{
"content": "If $a = \\log_8 225$ and $b = \\log_2 15$, then $a$, in terms of $b$, is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2b}{3}",
"style": "rule"
} | {
"index": "34311",
"split": "train"
} |
math | [
{
"content": "Melinda will roll two standard six-sided dice and make a two-digit number with the two numbers she rolls. For example, if she rolls a 6 and a 3, she can either form 36 or 63. What is the probability that she will be able to make an integer between 10 and 20, inclusive? Express your answer as a com... | {
"ground_truth": "\\frac{11}{36}",
"style": "rule"
} | {
"index": "34312",
"split": "train"
} |
math | [
{
"content": "John needs to catch a train. The train arrives randomly some time between 2:00 and 3:00, waits for 20 minutes, and then leaves. If John also arrives randomly between 2:00 and 3:00, what is the probability that the train will be there when John arrives? Please output the final answer within \\boxed... | {
"ground_truth": "\\frac{5}{18}",
"style": "rule"
} | {
"index": "34313",
"split": "train"
} |
math | [
{
"content": "Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must be painted? Please o... | {
"ground_truth": "876",
"style": "rule"
} | {
"index": "34314",
"split": "train"
} |
math | [
{
"content": "Sasha wrote the numbers $7, 8, 9, \\ldots, 17$ on the board and then erased one or more of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the maximum value that the sum of the rema... | {
"ground_truth": "121",
"style": "rule"
} | {
"index": "34315",
"split": "train"
} |
math | [
{
"content": "Bryan has some stamps of 3 cents, 4 cents, and 6 cents. What is the least number of stamps he can combine so the value of the stamps is 50 cents? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34316",
"split": "train"
} |
math | [
{
"content": "Let $x$ and $y$ be two-digit positive integers with mean $60$. What is the maximum value of the ratio $\\frac{x}{y}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{33}{7}",
"style": "rule"
} | {
"index": "34317",
"split": "train"
} |
math | [
{
"content": "How many integers between 1 and 200 are multiples of both 3 and 5 but not of either 4 or 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "34318",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ lies in the cartesian plane and has an area of $70$. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20),$ respectively, and the coordinates of $A$ are $(p,q).$ The line containing the median to side $BC$ has slope $-5.$ Find the largest possible value of $p+q.$\n[asy]defaultp... | {
"ground_truth": "47",
"style": "rule"
} | {
"index": "34319",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a quadrilateral with an inscribed circle, centre $O$ . Let \n\\[AO = 5, BO =6, CO = 7, DO = 8.\\] \nIf $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$ , determine $\\frac{OM}{ON}$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35/48",
"style": "rule"
} | {
"index": "34320",
"split": "train"
} |
math | [
{
"content": "Detached calculation.\n\n327 + 46 - 135\n\n1000 - 582 - 128\n\n(124 - 62) × 6\n\n500 - 400 ÷ 5 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "34321",
"split": "train"
} |
math | [
{
"content": "The sequence $\\left\\{a_{n}\\right\\}_{n \\geq 1}$ is defined by $a_{n+2}=7 a_{n+1}-a_{n}$ for positive integers $n$ with initial values $a_{1}=1$ and $a_{2}=8$. Another sequence, $\\left\\{b_{n}\\right\\}$, is defined by the rule $b_{n+2}=3 b_{n+1}-b_{n}$ for positive integers $n$ together with ... | {
"ground_truth": "89",
"style": "rule"
} | {
"index": "34322",
"split": "train"
} |
math | [
{
"content": "Let $m, n \\in \\mathbb{N}$, and $f(x) = (1+x)^m + (1+x)^n$.\n1. When $m=n=7$, $f(x) = a_7x^7 + a_6x^6 + \\ldots + a_1x + a_0$, find $a_0 + a_2 + a_4 + a_6$.\n2. If the coefficient of the expanded form of $f(x)$ is 19 when $m, n$ vary, find the minimum value of the coefficient of $x^2$. Please out... | {
"ground_truth": "81",
"style": "rule"
} | {
"index": "34323",
"split": "train"
} |
math | [
{
"content": "Consider a coordinate plane where at each lattice point, there is a circle with radius $\\frac{1}{8}$ and a square with sides of length $\\frac{1}{4}$, whose sides are parallel to the coordinate axes. A line segment runs from $(0,0)$ to $(729, 243)$. Determine how many of these squares and how man... | {
"ground_truth": "972",
"style": "rule"
} | {
"index": "34324",
"split": "train"
} |
math | [
{
"content": "Let\n\\[z = \\frac{-\\sqrt{3} + i}{2}.\\]Compute $z^6.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "34325",
"split": "train"
} |
math | [
{
"content": "Let \\[f(x) =\n\\begin{cases}\n3x + 5 &\\text{if }x<-3, \\\\\n7-4x&\\text{if }x\\ge -3.\n\\end{cases}\n\\]Find $f(5)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-13",
"style": "rule"
} | {
"index": "34326",
"split": "train"
} |
math | [
{
"content": "Given that Mr. A initially owns a home worth $\\$15,000$, he sells it to Mr. B at a $20\\%$ profit, then Mr. B sells it back to Mr. A at a $15\\%$ loss, then Mr. A sells it again to Mr. B at a $10\\%$ profit, and finally Mr. B sells it back to Mr. A at a $5\\%$ loss, calculate the net effect of th... | {
"ground_truth": "3541.50",
"style": "rule"
} | {
"index": "34327",
"split": "train"
} |
math | [
{
"content": "Form a four-digit number without repeating digits using the numbers 0, 1, 2, 3, 4, 5, 6, where the sum of the digits in the units, tens, and hundreds places is even. How many such four-digit numbers are there? (Answer with a number) Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "324",
"style": "rule"
} | {
"index": "34328",
"split": "train"
} |
math | [
{
"content": "To investigate the growth inhibitory effect of a certain drug on mice, $40$ mice were divided into two groups, a control group (without the drug) and an experimental group (with the drug).<br/>$(1)$ Suppose the number of mice in the control group among two mice is $X$, find the probability distrib... | {
"ground_truth": "95\\%",
"style": "rule"
} | {
"index": "34329",
"split": "train"
} |
math | [
{
"content": "If $\\frac{2+3+4}{3}=\\frac{1990+1991+1992}{N}$, then $N=$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1991",
"style": "rule"
} | {
"index": "34330",
"split": "train"
} |
math | [
{
"content": "Let $A = \\{x \\mid x^2 - ax + a^2 - 19 = 0\\}$, $B = \\{x \\mid x^2 - 5x + 6 = 0\\}$, and $C = \\{x \\mid x^2 + 2x - 8 = 0\\}$. \n(1) If $A = B$, find the value of $a$; \n(2) If $B \\cap A \\neq \\emptyset$ and $C \\cap A = \\emptyset$, find the value of $a$. Please output the final answer with... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "34331",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system on the plane, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar coordinate equation of the curve C₁ is ρ²-6ρcosθ+5=0, and the parametric equation of the curve C₂... | {
"ground_truth": "\\sqrt {7}",
"style": "rule"
} | {
"index": "34332",
"split": "train"
} |
math | [
{
"content": "Calculate the value of $({-\\frac{4}{5}})^{2022} \\times ({\\frac{5}{4}})^{2021}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "34333",
"split": "train"
} |
math | [
{
"content": "Given that \\(a_{1}, a_{2}, \\cdots, a_{n}\\) are \\(n\\) people corresponding to \\(A_{1}, A_{2}, \\cdots, A_{n}\\) cards (\\(n \\geq 2\\), \\(a_{i}\\) corresponds to \\(A_{i}\\)). Now \\(a_{1}\\) picks a card from the deck randomly, and then each person in sequence picks a card. If their corresp... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "34334",
"split": "train"
} |
math | [
{
"content": "Let $G, A_{1}, A_{2}, A_{3}, A_{4}, B_{1}, B_{2}, B_{3}, B_{4}, B_{5}$ be ten points on a circle such that $G A_{1} A_{2} A_{3} A_{4}$ is a regular pentagon and $G B_{1} B_{2} B_{3} B_{4} B_{5}$ is a regular hexagon, and $B_{1}$ lies on minor arc $G A_{1}$. Let $B_{5} B_{3}$ intersect $B_{1} A_{2}... | {
"ground_truth": "12^{\\circ}",
"style": "rule"
} | {
"index": "34335",
"split": "train"
} |
math | [
{
"content": "You are trying to cross a 400 foot wide river. You can jump at most 4 feet, but you have many stones you can throw into the river. You will stop throwing stones and cross the river once you have placed enough stones to be able to do so. You can throw straight, but you can't judge distance very wel... | {
"ground_truth": "712.811",
"style": "rule"
} | {
"index": "34336",
"split": "train"
} |
math | [
{
"content": "Given \\(0 \\leqslant x \\leqslant 2\\), the function \\(y=4^{x-\\frac{1}{2}}-3 \\cdot 2^{x}+5\\) reaches its minimum value at? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "34337",
"split": "train"
} |
math | [
{
"content": "A and B start from points A and B simultaneously, moving towards each other and meet at point C. If A starts 2 minutes earlier, then their meeting point is 42 meters away from point C. Given that A's speed is \\( a \\) meters per minute, B's speed is \\( b \\) meters per minute, where \\( a \\) an... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "34338",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer that can be written in the form $2002m + 44444n$, where $m$ and $n$ are integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "34339",
"split": "train"
} |
math | [
{
"content": "Given that the random variable $X$ follows a normal distribution $N(0,\\sigma^{2})$, if $P(X > 2) = 0.023$, determine the probability $P(-2 \\leqslant X \\leqslant 2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.954",
"style": "rule"
} | {
"index": "34340",
"split": "train"
} |
math | [
{
"content": "Granny Smith has $63. Elberta has $2 more than Anjou and Anjou has one-third as much as Granny Smith. How many dollars does Elberta have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "34341",
"split": "train"
} |
math | [
{
"content": "Maya owns 16 pairs of shoes, consisting of 8 identical black pairs, 4 identical brown pairs, 3 identical grey pairs, and 1 pair of white shoes. If Maya randomly picks two shoes, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe? Please o... | {
"ground_truth": "\\frac{45}{248}",
"style": "rule"
} | {
"index": "34342",
"split": "train"
} |
math | [
{
"content": "A factory produces a certain type of component, and the inspector randomly selects 16 of these components from the production line each day to measure their dimensions (in cm). The dimensions of the 16 components selected in one day are as follows:\n\n10.12, 9.97, 10.01, 9.95, 10.02, 9.98, 9.21, 1... | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "34343",
"split": "train"
} |
math | [
{
"content": "In the disaster relief donation, $\\frac{1}{10}$ of the people in a company each donated 200 yuan, $\\frac{3}{4}$ of the people each donated 100 yuan, and the remaining people each donated 50 yuan. Find the average donation per person in the company. Please output the final answer within \\boxed{}... | {
"ground_truth": "102.5",
"style": "rule"
} | {
"index": "34344",
"split": "train"
} |
math | [
{
"content": "A seven-digit natural number \\( N \\) is called interesting if:\n\n- It consists of non-zero digits;\n- It is divisible by 4;\n- Any number obtained from \\( N \\) by permuting its digits is also divisible by 4.\n\nHow many interesting numbers exist? Please output the final answer within \\boxed{... | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "34345",
"split": "train"
} |
math | [
{
"content": "Find the largest six-digit number in which all digits are distinct, and each digit, except the first and last ones, is either the sum or the difference of its neighboring digits. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "972538",
"style": "rule"
} | {
"index": "34346",
"split": "train"
} |
math | [
{
"content": "An integer $n$ is chosen uniformly at random from the set $\\{1,2,3, \\ldots, 2023!\\}$. Compute the probability that $$\\operatorname{gcd}\\left(n^{n}+50, n+1\\right)=1$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{265}{357}",
"style": "rule"
} | {
"index": "34347",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $\\angle A=45^{\\circ}$, $a=6$.\n(1) If $\\angle C=105^{\\circ}$, find $b$;\n(2) Find the maximum area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "9(1+\\sqrt{2})",
"style": "rule"
} | {
"index": "34348",
"split": "train"
} |
math | [
{
"content": "Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by $5$ minutes over the preceding day. Each of the four d... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "34349",
"split": "train"
} |
math | [
{
"content": "Square the numbers \\(a=1001\\) and \\(b=1001001\\). Extract the square root of the number \\(c=1002003004005004003002001\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1001001001001",
"style": "rule"
} | {
"index": "34350",
"split": "train"
} |
math | [
{
"content": "Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square then it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2\\times4$ rectangle ins... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34351",
"split": "train"
} |
math | [
{
"content": "Let \\( n \\) be a positive integer not exceeding 1996. If there exists a \\( \\theta \\) such that \\( (\\sin \\theta + i \\cos \\theta)^{n} = \\sin \\theta + i \\cos n \\theta \\), find the number of possible values for \\( n \\). Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "499",
"style": "rule"
} | {
"index": "34352",
"split": "train"
} |
math | [
{
"content": "Three balls are randomly placed into three boxes. Let the random variable $\\xi$ denote the maximum number of balls in any one box. Determine the mathematical expectation $E(\\xi)$ of $\\xi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17}{9}",
"style": "rule"
} | {
"index": "34353",
"split": "train"
} |
math | [
{
"content": "Given the wheel with a circumference of $11$ feet, the speed $r$ in miles per hour for which the time for a complete rotation of the wheel is shortened by $\\frac{1}{4}$ of a second is increased by $5$ miles per hour, find the value of $r$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34354",
"split": "train"
} |
math | [
{
"content": "Vasya has a stick that is 22 cm long. He wants to break it into three parts with integer lengths and form a triangle from the resulting pieces. How many ways can he do this? (Methods that result in identical triangles are considered the same). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34355",
"split": "train"
} |
math | [
{
"content": "Find the largest $K$ satisfying the following:\n\nGiven any closed intervals $A_1,\\ldots, A_N$ of length $1$ where $N$ is an arbitrary positive integer. If their union is $[0,2021]$ , then we can always find $K$ intervals from $A_1,\\ldots, A_N$ such that the intersection of any two ... | {
"ground_truth": "1011",
"style": "rule"
} | {
"index": "34356",
"split": "train"
} |
math | [
{
"content": "Let $S_n$ and $T_n$ respectively be the sum of the first $n$ terms of the arithmetic sequences $\\{a_n\\}$ and $\\{b_n\\}$. Given that $\\frac{S_n}{T_n} = \\frac{n}{2n+1}$ for $n \\in \\mathbb{N}^*$, find the value of $\\frac{a_5}{b_5}$. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "\\frac{9}{19}",
"style": "rule"
} | {
"index": "34357",
"split": "train"
} |
math | [
{
"content": "Jeffrey jogs $3$ miles at $4$ miles per hour. Then, he walks $4$ miles at $3$ miles an hour. What was Jeffrey's average speed in miles per hour for the entire trip? Express your answer as a decimal to the nearest hundredth. Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "3.36",
"style": "rule"
} | {
"index": "34358",
"split": "train"
} |
math | [
{
"content": "A circle touches the sides of an angle at points \\( A \\) and \\( B \\). The distance from a point \\( C \\) lying on the circle to the line \\( AB \\) is 6. Find the sum of the distances from point \\( C \\) to the sides of the angle, given that one of these distances is nine times less than the... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34359",
"split": "train"
} |
math | [
{
"content": "An integer $n>1$ is given . Find the smallest positive number $m$ satisfying the following conditions： for any set $\\{a,b\\}$ $\\subset \\{1,2,\\cdots,2n-1\\}$ ,there are non-negative integers $ x, y$ ( not all zero) such that $2n|ax+by$ and $x+y\\leq m.$ Please output the final answer w... | {
"ground_truth": "n",
"style": "rule"
} | {
"index": "34360",
"split": "train"
} |
math | [
{
"content": "How many positive integers $2 \\leq a \\leq 101$ have the property that there exists a positive integer $N$ for which the last two digits in the decimal representation of $a^{2^{n}}$ is the same for all $n \\geq N$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "34361",
"split": "train"
} |
math | [
{
"content": "Fifty cards, numbered 1-50, are placed in a box. One card is randomly selected. What is the probability that the number on the card is prime and is a multiple of 7? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{50}",
"style": "rule"
} | {
"index": "34362",
"split": "train"
} |
math | [
{
"content": "How many solutions in nonnegative integers $(a, b, c)$ are there to the equation $2^{a}+2^{b}=c!\\quad ?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "34363",
"split": "train"
} |
math | [
{
"content": "In the coordinate plane, points $D(3, 15)$, $E(15, 0)$, and $F(0, t)$ form a triangle $\\triangle DEF$. If the area of $\\triangle DEF$ is 50, what is the value of $t$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{125}{12}",
"style": "rule"
} | {
"index": "34364",
"split": "train"
} |
math | [
{
"content": "A zoo has six pairs of different animals (one male and one female for each species). The zookeeper wishes to begin feeding the male lion and must alternate between genders each time. Additionally, after feeding any lion, the next animal cannot be a tiger. How many ways can he feed all the animals?... | {
"ground_truth": "14400",
"style": "rule"
} | {
"index": "34365",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, point $E$ is on $AB$, point $F$ is on $AC$, and $BF$ intersects $CE$ at point $P$. If the areas of quadrilateral $AEPF$ and triangles $BEP$ and $CFP$ are all equal to 4, what is the area of $\\triangle BPC$? Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34366",
"split": "train"
} |
math | [
{
"content": "Given a circle that is tangent to the side \\(DC\\) of a regular pentagon \\(ABCDE\\) at point \\(D\\) and tangent to the side \\(AB\\) at point \\(A\\), what is the degree measure of the minor arc \\(AD\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "34367",
"split": "train"
} |
math | [
{
"content": "There are births in West Northland every 6 hours, deaths every 2 days, and a net immigration every 3 days. Calculate the approximate annual increase in population. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1400",
"style": "rule"
} | {
"index": "34368",
"split": "train"
} |
math | [
{
"content": "What is the sum of the digits of the decimal representation of $2^{2005} \\times 5^{2007} \\times 3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34369",
"split": "train"
} |
math | [
{
"content": "15 plus 16 equals. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "34370",
"split": "train"
} |
math | [
{
"content": "Initially, the fairy tale island was divided into three counties: in the first county lived only elves, in the second - only dwarves, and in the third - only centaurs.\n\n- During the first year, each county where there were no elves was divided into three counties.\n- During the second year, each... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "34371",
"split": "train"
} |
math | [
{
"content": "A hotpot restaurant in Chongqing operates through three methods: dining in, takeout, and setting up a stall outside (referred to as stall). In June, the ratio of revenue from dining in, takeout, and stall for this hotpot restaurant was $3:5:2$. With the introduction of policies to promote consumpt... | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "34372",
"split": "train"
} |
math | [
{
"content": "In $\\triangle BAC$, $\\angle BAC=40^\\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\\overline{AB}$ and $\\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "34373",
"split": "train"
} |
math | [
{
"content": "In a new sequence, the first term is \\(a_1 = 5000\\) and the second term is \\(a_2 = 5001\\). Furthermore, the values of the remaining terms are designed so that \\(a_n + a_{n+1} + a_{n+2} = 2n\\) for all \\( n \\geq 1 \\). Determine \\(a_{1000}\\). Please output the final answer within \\boxed{}... | {
"ground_truth": "5666",
"style": "rule"
} | {
"index": "34374",
"split": "train"
} |
math | [
{
"content": "Suppose a sequence of positive numbers $\\left\\{a_{n}\\right\\}$ satisfies: $a_{0} = 1, a_{n} = a_{n+1} + a_{n+2},$ for $n = 0, 1, 2, \\ldots$. Find $a_{1}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}",
"style": "rule"
} | {
"index": "34375",
"split": "train"
} |
math | [
{
"content": "Determine the greatest integer less than or equal to \\[\\frac{5^{50} + 3^{50}}{5^{45} + 3^{45}}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3124",
"style": "rule"
} | {
"index": "34376",
"split": "train"
} |
math | [
{
"content": "There are $10$ horses, named Horse 1, Horse 2, $\\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The hor... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "34377",
"split": "train"
} |
math | [
{
"content": "When $10^{93}-93$ is expressed as a single whole number, the sum of the digits is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "826",
"style": "rule"
} | {
"index": "34378",
"split": "train"
} |
math | [
{
"content": "For what base-6 digit $d$ is $2dd5_6$ divisible by the base 10 number 11? (Here $2dd5_6$ represents a base-6 number whose first digit is 2, whose last digit is 5, and whose middle two digits are both equal to $d$). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "34379",
"split": "train"
} |
math | [
{
"content": "A group consists of 4 boys and 3 girls.\n(1) If one boy and one girl are each chosen to host a program, how many different selections are possible?\n(2) If two boys and two girls are chosen to form a small choir, and are to be arranged in a row with the requirement that the two girls are not adjac... | {
"ground_truth": "216",
"style": "rule"
} | {
"index": "34380",
"split": "train"
} |
math | [
{
"content": "Suppose that $m$ and $n$ are positive integers with $\\sqrt{7+\\sqrt{48}}=m+\\sqrt{n}$. What is the value of $m^{2}+n^{2}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "34381",
"split": "train"
} |
math | [
{
"content": "Given that the focus of the parabola $y^{2}=8x$ is $F$, two perpendicular lines $l_{1}$ and $l_{2}$ are drawn passing through point $F$. Let these lines intersect the parabola at points $A$, $B$ and $D$, $E$, respectively. The minimum area of quadrilateral $ADBE$ is ____. Please output the final a... | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "34382",
"split": "train"
} |
math | [
{
"content": "A right triangle with legs each measuring 7 cells was drawn on graph paper. Then all the lines of the grid inside the triangle were outlined. What is the maximum number of triangles that can be found in this drawing? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "34383",
"split": "train"
} |
math | [
{
"content": "The length of the longer side of rectangle \\(R'\\) is \\(15\\%\\) more than the length of a side of square \\(S'\\). The length of the shorter side of rectangle \\(R'\\) is \\(5\\%\\) less than the length of a side of square \\(S'\\). What is the ratio of the area of rectangle \\(R'\\) to the are... | {
"ground_truth": "\\frac{10925}{10000}",
"style": "rule"
} | {
"index": "34384",
"split": "train"
} |
math | [
{
"content": "The side length of square \\(ABCD\\) is 4. Point \\(E\\) is the midpoint of \\(AB\\), and point \\(F\\) is a moving point on side \\(BC\\). Triangles \\(\\triangle ADE\\) and \\(\\triangle DCF\\) are folded up along \\(DE\\) and \\(DF\\) respectively, making points \\(A\\) and \\(C\\) coincide at ... | {
"ground_truth": "\\frac{4\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "34385",
"split": "train"
} |
math | [
{
"content": "The students in Mrs. Reed's English class are reading the same $760$-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in $20$ seconds, Bob reads a page in $45$ seconds and Chandra reads a page in $30$ seconds. Chandra and Bob, who each have a copy of the bo... | {
"ground_truth": "456",
"style": "rule"
} | {
"index": "34386",
"split": "train"
} |
math | [
{
"content": "Given that the radius of a sphere is 24cm, and the height of a cone is equal to the diameter of this sphere, and the surface area of the sphere is equal to the surface area of the cone, then the volume of this cone is \\_\\_\\_\\_\\_\\_ cm<sup>3</sup>. Please output the final answer within \\boxed... | {
"ground_truth": "12288\\pi",
"style": "rule"
} | {
"index": "34387",
"split": "train"
} |
math | [
{
"content": "A two-digit number, when multiplied by 109, yields a four-digit number. It is divisible by 23, and the quotient is a one-digit number. The maximum value of this two-digit number is $\\qquad$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "69",
"style": "rule"
} | {
"index": "34388",
"split": "train"
} |
math | [
{
"content": "Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-64",
"style": "rule"
} | {
"index": "34389",
"split": "train"
} |
math | [
{
"content": "We have a standard deck of 52 cards, with 4 cards in each of 13 ranks. We call a 5-card poker hand a full house if the hand has 3 cards of one rank and 2 cards of another rank (such as 33355 or AAAKK). What is the probability that five cards chosen at random form a full house? Please output the ... | {
"ground_truth": "\\frac{6}{4165}",
"style": "rule"
} | {
"index": "34390",
"split": "train"
} |
math | [
{
"content": "The numbers \\( x \\) and \\( y \\) satisfy the equation:\n\n$$\n\\sqrt{x y}+\\sqrt{(1-x)(1-y)}=\\sqrt{7 x(1-y)}+\\frac{\\sqrt{y(1-x)}}{\\sqrt{7}}\n$$\n\nFind the maximum value of the expression \\( x + 7y \\). Justify your answer. Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "57/8",
"style": "rule"
} | {
"index": "34391",
"split": "train"
} |
math | [
{
"content": "Given $a > 1$, $b > 1$, and $$\\frac {1}{a-1} + \\frac {1}{b-1} = 1$$, find the minimum value of $a + 4b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "34392",
"split": "train"
} |
math | [
{
"content": "Each day Walter gets $3$ dollars for doing his chores or $5$ dollars for doing them exceptionally well. After $10$ days of doing his chores daily, Walter has received a total of $36$ dollars. On how many days did Walter do them exceptionally well? Please output the final answer within \\boxed{}.",... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "34393",
"split": "train"
} |
math | [
{
"content": "In a football tournament, each team is supposed to play one match against each of the other teams. However, during the tournament, half of the teams were disqualified and did not participate further. As a result, a total of 77 matches were played, and the disqualified teams managed to play all the... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "34394",
"split": "train"
} |
math | [
{
"content": "Find the remainder when $5^{5^{5^5}}$ is divided by 500. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "34395",
"split": "train"
} |
math | [
{
"content": "Each of two sides of a triangle is divided into six equal parts; the corresponding division points are connected by line segments. Find the sum of the lengths of these segments if the third side of the triangle is equal to 12. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "34396",
"split": "train"
} |
math | [
{
"content": "The sides of triangle $DEF$ are in the ratio of $3:4:5$. Segment $EG$ is the angle bisector drawn to the shortest side, dividing it into segments $DG$ and $GE$. What is the length, in inches, of the longer subsegment of side $DE$ if the length of side $DE$ is $12$ inches? Express your answer as a ... | {
"ground_truth": "\\frac{48}{7}",
"style": "rule"
} | {
"index": "34397",
"split": "train"
} |
math | [
{
"content": "The constant term in the expansion of \\\\(\\left(x^{2}- \\frac{1}{x}+3\\right)^{4}\\\\) is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "117",
"style": "rule"
} | {
"index": "34398",
"split": "train"
} |
math | [
{
"content": "A circle with a radius of 2 is inscribed in triangle \\(ABC\\) and touches side \\(BC\\) at point \\(D\\). Another circle with a radius of 4 touches the extensions of sides \\(AB\\) and \\(AC\\), as well as side \\(BC\\) at point \\(E\\). Find the area of triangle \\(ABC\\) if the measure of angle... | {
"ground_truth": "\\frac{56}{\\sqrt{3}}",
"style": "rule"
} | {
"index": "34399",
"split": "train"
} |
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