data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Let $N$ be a three-digit integer such that the difference between any two positive integer factors of $N$ is divisible by 3 . Let $d(N)$ denote the number of positive integers which divide $N$. Find the maximum possible value of $N \\cdot d(N)$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "5586",
"style": "rule"
} | {
"index": "7700",
"split": "train"
} |
math | [
{
"content": "For $x > 0$, find the maximum value of $f(x) = 1 - 2x - \\frac{3}{x}$ and the value of $x$ at which it occurs. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{6}}{2}",
"style": "rule"
} | {
"index": "7701",
"split": "train"
} |
math | [
{
"content": "Calculate $(-1)^{47} + 2^{(3^3+4^2-6^2)}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "127",
"style": "rule"
} | {
"index": "7702",
"split": "train"
} |
math | [
{
"content": "Let $\\alpha$ be an acute angle, and $\\cos\\left(\\alpha+ \\frac{\\pi}{6}\\right) = \\frac{3 \\sqrt{10}}{10}$, $\\tan(\\alpha+\\beta) = \\frac{2}{5}$. \n(1) Find the value of $\\sin\\left(2\\alpha+ \\frac{\\pi}{6}\\right)$. \n(2) Find the value of $\\tan\\left(2\\beta- \\frac{\\pi}{3}\\right)$.... | {
"ground_truth": "\\frac{17}{144}",
"style": "rule"
} | {
"index": "7703",
"split": "train"
} |
math | [
{
"content": "Let \\( P, Q, R, \\) and \\( S \\) be positive real numbers such that\n\\[\n\\log_{10} (PQ) + \\log_{10} (PS) = 2, \\\\\n\\log_{10} (QS) + \\log_{10} (QR) = 3, \\\\\n\\log_{10} (RP) + \\log_{10} (RS) = 5.\n\\]\nCompute the value of the product \\( PQRS \\). Please output the final answer within \\... | {
"ground_truth": "100000",
"style": "rule"
} | {
"index": "7704",
"split": "train"
} |
math | [
{
"content": "A younger brother leaves home and walks to the park at a speed of 4 kilometers per hour. Two hours later, the older brother leaves home and rides a bicycle at a speed of 20 kilometers per hour to catch up with the younger brother. How long will it take for the older brother to catch up with the yo... | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "7705",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. The altitude from $A$ intersects $B C$ at $D$. Let $\\omega_{1}$ and $\\omega_{2}$ be the incircles of $A B D$ and $A C D$, and let the common external tangent of $\\omega_{1}$ and $\\omega_{2}$ (other than $B C$) intersect $A D$ at $E$. Com... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "7706",
"split": "train"
} |
math | [
{
"content": "Four fair dice are tossed at random. What is the probability that the four numbers on the dice can be arranged to form an arithmetic progression with a common difference of one? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{432}",
"style": "rule"
} | {
"index": "7707",
"split": "train"
} |
math | [
{
"content": "Given $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, and $\\overrightarrow{a}⊥(\\overrightarrow{a}+\\overrightarrow{b})$, find the projection of $\\overrightarrow{a}$ onto $\\overrightarrow{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "7708",
"split": "train"
} |
math | [
{
"content": "Given that the height of a cylinder is $2$, and the circumferences of its two bases lie on the surface of the same sphere with a diameter of $2\\sqrt{6}$, calculate the surface area of the cylinder. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(10+4\\sqrt{5})\\pi",
"style": "rule"
} | {
"index": "7709",
"split": "train"
} |
math | [
{
"content": "Given that $|\\vec{a}| = 1$ and $\\vec{b} = (1, \\sqrt{3})$, and that $(\\vec{b} - \\vec{a}) \\perp \\vec{a}$, find the angle between vector $\\vec{a}$ and vector $\\vec{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "7710",
"split": "train"
} |
math | [
{
"content": "Let $P$ be a moving point on the line $3x+4y+3=0$, and through point $P$, draw two tangents to the circle $C$: $x^{2}+y^{2}-2x-2y+1=0$, with the points of tangency being $A$ and $B$, respectively. Find the minimum value of the area of quadrilateral $PACB$. Please output the final answer within \\b... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "7711",
"split": "train"
} |
math | [
{
"content": "Find the square root of $\\dfrac{10!}{210}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72\\sqrt{5}",
"style": "rule"
} | {
"index": "7712",
"split": "train"
} |
math | [
{
"content": "If $\\frac{b}{a} = 2$ and $\\frac{c}{b} = 3$, what is the ratio of $a + b$ to $b + c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{8}",
"style": "rule"
} | {
"index": "7713",
"split": "train"
} |
math | [
{
"content": "Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, \"I can't tell who has the larger number.\" Then Bob says, \"I know who has the larger number.\" Alice says,... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "7714",
"split": "train"
} |
math | [
{
"content": "What is the largest $n$ for which it is possible to construct two bi-infinite sequences $A$ and $B$ such that any subsequence of $B$ of length $n$ is contained in $A$, $A$ has a period of 1995, and $B$ does not have this property (is either non-periodic or has a period of a different length)? Plea... | {
"ground_truth": "1995",
"style": "rule"
} | {
"index": "7715",
"split": "train"
} |
math | [
{
"content": "Three flower beds, X, Y, and Z, contain X = 600, Y = 480, and Z = 420 plants, respectively. The plants are shared as follows: Beds X and Y share 60 plants, Beds Y and Z share 70 plants, and Beds X and Z share 80 plants. Also, there are 30 plants common to all three beds. Find the total number of u... | {
"ground_truth": "1320",
"style": "rule"
} | {
"index": "7716",
"split": "train"
} |
math | [
{
"content": "Find the smallest fraction which, when divided by each of the fractions \\(\\frac{21}{25}\\) and \\(\\frac{14}{15}\\), results in natural numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42/5",
"style": "rule"
} | {
"index": "7717",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy: $|\\overrightarrow{a}| = \\sqrt{2}$, $|\\overrightarrow{b}| = 2$, and $(\\overrightarrow{a} - \\overrightarrow{b}) \\perp \\overrightarrow{a}$, find the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Please... | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "7718",
"split": "train"
} |
math | [
{
"content": "Given that Mary reversed the digits of a two-digit multiplier $b$, while multiplying correctly with a positive integer $a$, and her erroneous product was $180$, determine the correct product of $a$ and $b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "180",
"style": "rule"
} | {
"index": "7719",
"split": "train"
} |
math | [
{
"content": "Twelve students in Mrs. Stephenson's class have brown eyes. Twenty students in the class have a lunch box. Of Mrs. Stephenson's 30 students, what is the least possible number of students who have brown eyes and a lunch box? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "7720",
"split": "train"
} |
math | [
{
"content": "Let $A,B,C$ be angles of an acute triangle with \\begin{align*} \\cos^2 A + \\cos^2 B + 2 \\sin A \\sin B \\cos C &= \\frac{15}{8} \\text{ and} \\\\ \\cos^2 B + \\cos^2 C + 2 \\sin B \\sin C \\cos A &= \\frac{14}{9} \\end{align*} There are positive integers $p$, $q$, $r$, and $s$ for which \\[\\co... | {
"ground_truth": "222",
"style": "rule"
} | {
"index": "7721",
"split": "train"
} |
math | [
{
"content": "Let \\(a_{1}, a_{2}, \\cdots, a_{k}\\) be a finite arithmetic sequence, such that \\(a_{4} + a_{7} + a_{10} = 17\\), and \\(a_{4} + a_{5} + a_{6} + \\cdots + a_{14} = 77\\), and \\(a_{k} = 13\\). Calculate the value of \\(k\\). Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "7722",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than 200 are divisible by 2, 3 and 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "7723",
"split": "train"
} |
math | [
{
"content": "A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery a... | {
"ground_truth": "351",
"style": "rule"
} | {
"index": "7724",
"split": "train"
} |
math | [
{
"content": "The median of the numbers 3, 7, x, 14, 20 is equal to the mean of those five numbers. Calculate the sum of all real numbers \\( x \\) for which this is true. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "7725",
"split": "train"
} |
math | [
{
"content": "Find the value of \\(\\log _{2}\\left[2^{3} 4^{4} 8^{5} \\cdots\\left(2^{20}\\right)^{22}\\right]\\). \nChoose one of the following options:\n(a) 3290\n(b) 3500\n(c) 3710\n(d) 4172 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5950",
"style": "rule"
} | {
"index": "7726",
"split": "train"
} |
math | [
{
"content": "There are 7 cards, each with a number written on it: 1, 2, 2, 3, 4, 5, 6. Three cards are randomly drawn from these 7 cards. Let the minimum value of the numbers on the drawn cards be $\\xi$. Find $P\\left(\\xi =2\\right)=$____ and $E\\left(\\xi \\right)=$____. Please output the final answer withi... | {
"ground_truth": "\\frac{12}{7}",
"style": "rule"
} | {
"index": "7727",
"split": "train"
} |
math | [
{
"content": "In a quadrilateral $ABCD$ lying in the plane, $AB=\\sqrt{3}$, $AD=DC=CB=1$. The areas of triangles $ABD$ and $BCD$ are $S$ and $T$ respectively. What is the maximum value of $S^{2} + T^{2}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{8}",
"style": "rule"
} | {
"index": "7728",
"split": "train"
} |
math | [
{
"content": "Given $f(x) = 4\\cos x\\sin \\left(x+ \\frac{\\pi}{6}\\right)-1$.\n(Ⅰ) Determine the smallest positive period of $f(x)$;\n(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $\\left[- \\frac{\\pi}{6}, \\frac{\\pi}{4}\\right]$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "7729",
"split": "train"
} |
math | [
{
"content": "Let $z$ be a complex number satisfying $z^2 = 4z - 19 + 8i$. Given that $|z|$ is an integer, find $z.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 + 4i",
"style": "rule"
} | {
"index": "7730",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered pairs $(a,b)$ of complex numbers such that\n\\[a^3 b^5 = a^7 b^2 = 1.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "7731",
"split": "train"
} |
math | [
{
"content": "Calculate: $\\sqrt{27} - 2\\cos 30^{\\circ} + \\left(\\frac{1}{2}\\right)^{-2} - |1 - \\sqrt{3}|$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3} + 5",
"style": "rule"
} | {
"index": "7732",
"split": "train"
} |
math | [
{
"content": "Suppose that a parabola has vertex $\\left(\\frac{1}{4},-\\frac{9}{8}\\right)$ and equation $y = ax^2 + bx + c$, where $a > 0$ and $a + b + c$ is an integer. The minimum possible value of $a$ can be written in the form $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find ... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "7733",
"split": "train"
} |
math | [
{
"content": "Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. T... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "7734",
"split": "train"
} |
math | [
{
"content": "What is the value of $\\sqrt{2 \\cdot 4! \\cdot 4!}$ expressed as a positive integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24\\sqrt{2}",
"style": "rule"
} | {
"index": "7735",
"split": "train"
} |
math | [
{
"content": "You have a rectangular prism box with length $x+5$ units, width $x-5$ units, and height $x^{2}+25$ units. For how many positive integer values of $x$ is the volume of the box less than 700 units? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "7736",
"split": "train"
} |
math | [
{
"content": "A math teacher randomly selects 3 questions for analysis from a test paper consisting of 12 multiple-choice questions, 4 fill-in-the-blank questions, and 6 open-ended questions. The number of different ways to select questions such that at least one multiple-choice question and at least one open-e... | {
"ground_truth": "864",
"style": "rule"
} | {
"index": "7737",
"split": "train"
} |
math | [
{
"content": "$ABCD$ is a regular tetrahedron. Let $N$ be the midpoint of $\\overline{AB}$, and let $G$ be the centroid of triangle $ACD$. What is $\\cos \\angle BNG$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "7738",
"split": "train"
} |
math | [
{
"content": "Given $f(x) = 2\\sin x (\\sin x + \\cos x)$, where $x \\in \\mathbb{R}$.\n\n(Ⅰ) Find the intervals of monotonic increase for the function $f(x)$.\n\n(Ⅱ) If $f\\left( \\frac{a}{2} \\right) = 1 + \\frac{3\\sqrt{2}}{5}$, and $\\frac{3\\pi}{4} < a < \\frac{5\\pi}{4}$, find the value of $\\cos a$. Plea... | {
"ground_truth": "-\\frac{7\\sqrt{2}}{10}",
"style": "rule"
} | {
"index": "7739",
"split": "train"
} |
math | [
{
"content": "What is the maximum value that can be taken by the sum\n\n$$\n\\left|x_{1}-1\\right|+\\left|x_{2}-2\\right|+\\ldots+\\left|x_{63}-63\\right|\n$$\n\nif $x_{1}, x_{2}, \\ldots, x_{63}$ are some permutation of the numbers $1, 2, 3, \\ldots, 63$? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "1984",
"style": "rule"
} | {
"index": "7740",
"split": "train"
} |
math | [
{
"content": "The four partners in a business decide to split the profits of their company in the ratio 2:3:3:5. If the profit one year is $\\$26,\\!000$ what is the largest number of dollars received by any of the four partners? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10,\\!000",
"style": "rule"
} | {
"index": "7741",
"split": "train"
} |
math | [
{
"content": "Inside a right circular cone with base radius $8$ and height $15$, there are three identical spheres. Each sphere is tangent to the others, the base, and the side of the cone. Determine the radius $r$ of each sphere.\nA) $\\frac{840 - 300\\sqrt{3}}{121}$\nB) $\\frac{60}{19 + 5\\sqrt{3}}$\nC) $\\fr... | {
"ground_truth": "\\frac{280 - 100\\sqrt{3}}{121}",
"style": "rule"
} | {
"index": "7742",
"split": "train"
} |
math | [
{
"content": "For every real number $x$, what is the value of the expression $(x+1)^{2} - x^{2}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2x + 1",
"style": "rule"
} | {
"index": "7743",
"split": "train"
} |
math | [
{
"content": "We wrote the reciprocals of natural numbers from 2 to 2011 on a board. In one step, we erase two numbers, \\( x \\) and \\( y \\), and replace them with the number\n\n$$\n\\frac{xy}{xy + (1 - x)(1 - y)}\n$$\n\nBy repeating this process 2009 times, only one number remains. What could this number be... | {
"ground_truth": "\\frac{1}{2010! + 1}",
"style": "rule"
} | {
"index": "7744",
"split": "train"
} |
math | [
{
"content": "A store increased the price of a certain Super VCD by 40% and then advertised a \"10% discount and a free 50 yuan taxi fare\" promotion. As a result, each Super VCD still made a profit of 340 yuan. What was the cost price of each Super VCD? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "1500",
"style": "rule"
} | {
"index": "7745",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}=( \\sqrt {3}\\sin x,m+\\cos x)$ and $\\overrightarrow{b}=(\\cos x,-m+\\cos x)$, and the function $f(x)= \\overrightarrow{a}\\cdot \\overrightarrow{b}$\n(1) Find the analytical expression for the function $f(x)$;\n(2) When $x\\in\\left[-\\frac{\\pi}{6}, \\frac{\\p... | {
"ground_truth": "-\\frac{3}{2}",
"style": "rule"
} | {
"index": "7746",
"split": "train"
} |
math | [
{
"content": "How many 8-digit numbers begin with 1 , end with 3 , and have the property that each successive digit is either one more or two more than the previous digit, considering 0 to be one more than 9 ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "7747",
"split": "train"
} |
math | [
{
"content": "Let $x_1, x_2, ... , x_6$ be non-negative real numbers such that $x_1 +x_2 +x_3 +x_4 +x_5 +x_6 =1$, and $x_1 x_3 x_5 +x_2 x_4 x_6 \\ge {\\frac{1}{540}}$. Let $p$ and $q$ be positive relatively prime integers such that $\\frac{p}{q}$ is the maximum possible value of $x_1 x_2 x_3 + x_2 x_3 x_4 +x_3 ... | {
"ground_truth": "559",
"style": "rule"
} | {
"index": "7748",
"split": "train"
} |
math | [
{
"content": "A biologist sequentially placed 150 beetles into ten jars. In each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is at least half of the number of beetles in the tenth jar. How many beetles are in the sixth jar? Please output the final answ... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "7749",
"split": "train"
} |
math | [
{
"content": "Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (list... | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "7750",
"split": "train"
} |
math | [
{
"content": "The average age of 8 people in a room is 35 years. A 22-year-old person leaves the room. Calculate the average age of the seven remaining people. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{258}{7}",
"style": "rule"
} | {
"index": "7751",
"split": "train"
} |
math | [
{
"content": "Let $O$ and $H$ denote the circumcenter and orthocenter of triangle $ABC,$ respectively. If $AO = AH,$ then enter all possible values of $\\angle A$ (in degrees), separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60^\\circ, 120^\\circ",
"style": "rule"
} | {
"index": "7752",
"split": "train"
} |
math | [
{
"content": "In the trapezoid \\( ABCD \\), angles \\( A \\) and \\( D \\) are right angles, \\( AB = 1 \\), \\( CD = 4 \\), \\( AD = 5 \\). Point \\( M \\) is taken on side \\( AD \\) such that \\(\\angle CMD = 2 \\angle BMA \\).\n\nIn what ratio does point \\( M \\) divide side \\( AD \\)? Please output the ... | {
"ground_truth": "2:3",
"style": "rule"
} | {
"index": "7753",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers that satisfy \\[2\\log_{x}(2y) = 2\\log_{2x}(4z) = \\log_{2x^4}(8yz) \\ne 0.\\] The value of $xy^5z$ can be expressed in the form $\\frac{1}{2^{p/q}},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$ Please output the final answe... | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "7754",
"split": "train"
} |
math | [
{
"content": "What is $2a+3b$, where $a=2-i$ and $b=-1+i$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1+i",
"style": "rule"
} | {
"index": "7755",
"split": "train"
} |
math | [
{
"content": "How many positive odd integers greater than 1 and less than $200$ are square-free? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "82",
"style": "rule"
} | {
"index": "7756",
"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 $\\diamondsuit$) are red, and the other two ($\\spadesuit$ and $\\clubsuit$) are black. The cards in the deck are shuffled. What is the probability that the first card drawn ... | {
"ground_truth": "\\frac{169}{2652}",
"style": "rule"
} | {
"index": "7757",
"split": "train"
} |
math | [
{
"content": "Simplify $\\frac{180}{16}\\cdot \\frac{5}{120}\\cdot \\frac{8}{3}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "7758",
"split": "train"
} |
math | [
{
"content": "Little Tiger places chess pieces on the grid points of a 19 × 19 Go board, forming a solid rectangular dot matrix. Then, by adding 45 more chess pieces, he transforms it into a larger solid rectangular dot matrix with one side unchanged. What is the maximum number of chess pieces that Little Tiger... | {
"ground_truth": "285",
"style": "rule"
} | {
"index": "7759",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be a triangle with $A B=9, B C=10$, and $C A=17$. Let $B^{\\prime}$ be the reflection of the point $B$ over the line $C A$. Let $G$ be the centroid of triangle $A B C$, and let $G^{\\prime}$ be the centroid of triangle $A B^{\\prime} C$. Determine the length of segment $G G^{\\prime}$.... | {
"ground_truth": "\\frac{48}{17}",
"style": "rule"
} | {
"index": "7760",
"split": "train"
} |
math | [
{
"content": "Any five points are taken inside or on a square with side length $1$. Let a be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than $a$. Then $a$ is:\n$\\textbf{(A)}\\ \\s... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "7761",
"split": "train"
} |
math | [
{
"content": "Let $a_1=24$ and form the sequence $a_n$ , $n\\geq 2$ by $a_n=100a_{n-1}+134$ . The first few terms are $$ 24,2534,253534,25353534,\\ldots $$ What is the least value of $n$ for which $a_n$ is divisible by $99$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "88",
"style": "rule"
} | {
"index": "7762",
"split": "train"
} |
math | [
{
"content": "When three different numbers from the set $\\{ -3, -2, -1, 4, 5 \\}$ are multiplied, the largest possible product is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "7763",
"split": "train"
} |
math | [
{
"content": "From noon till midnight, Clever Cat sleeps under the oak tree and from midnight till noon he is awake telling stories. A poster on the tree above him says \"Two hours ago, Clever Cat was doing the same thing as he will be doing in one hour's time\". For how many hours a day does the poster tell th... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "7764",
"split": "train"
} |
math | [
{
"content": "The expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $1:2:3$ that can be written in the form\\[{n\\choose k} : {n\\choose k+1} : {n \\choose k+2}\\]Find the sum of all possible values of $n+k$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "7765",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $n$ such that, if there are initially $n+1$ townspeople and $n$ goons, then the probability the townspeople win is less than $1\\%$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "7766",
"split": "train"
} |
math | [
{
"content": "Tom, John, and Lily each shot six arrows at a target. Arrows hitting anywhere within the same ring scored the same number of points. Tom scored 46 points and John scored 34 points. How many points did Lily score? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "7767",
"split": "train"
} |
math | [
{
"content": "In writing the integers from 10 through 99 inclusive, how many times is the digit 7 written? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "7768",
"split": "train"
} |
math | [
{
"content": "If the sum $1 + 2 + 3 + \\cdots + K$ is a perfect square $N^2$ and if $N$ is less than $100$, then the possible values for $K$ are: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1, 8, and 49",
"style": "rule"
} | {
"index": "7769",
"split": "train"
} |
math | [
{
"content": "Every second, Andrea writes down a random digit uniformly chosen from the set $\\{1,2,3,4\\}$. She stops when the last two numbers she has written sum to a prime number. What is the probability that the last number she writes down is 1? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "15/44",
"style": "rule"
} | {
"index": "7770",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC,$ $D$ is on $\\overline{AB}$ such that $AD:DB = 3:2,$ and $E$ is on $\\overline{BC}$ such that $BE:EC = 3:2.$ If lines $DE$ and $AC$ intersect at $F,$ then find $\\frac{DE}{EF}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "7771",
"split": "train"
} |
math | [
{
"content": "Let $f(x)$ be a monic quartic polynomial such that $f(-2)=-4$, $f(1)=-1$, $f(-3)=-9$, and $f(5)=-25$. Find $f(2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-64",
"style": "rule"
} | {
"index": "7772",
"split": "train"
} |
math | [
{
"content": "Henry constructs a model involving four spherical clay pieces of radii 1 inch, 4 inches, 6 inches, and 3 inches. Calculate the square of the total volume of the clay used, subtracting 452 from the squared result, and express your answer in terms of $\\pi$. Please output the final answer within \\b... | {
"ground_truth": "168195.11\\pi^2",
"style": "rule"
} | {
"index": "7773",
"split": "train"
} |
math | [
{
"content": "What is the value of the expression $[ a-(b-c) ] - [(a-b) - c ]$ when $a = 17$, $b=21$ and $c=5$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "7774",
"split": "train"
} |
math | [
{
"content": "A particular coin has a $\\frac{1}{3}$ chance of landing on heads (H), $\\frac{1}{3}$ chance of landing on tails (T), and $\\frac{1}{3}$ chance of landing vertically in the middle (M). When continuously flipping this coin, what is the probability of observing the continuous sequence HMMT before HM... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "7775",
"split": "train"
} |
math | [
{
"content": "The inclination angle of the line $x-y+1=0$ can be calculated. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "7776",
"split": "train"
} |
math | [
{
"content": "In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "7777",
"split": "train"
} |
math | [
{
"content": "In the equation \"Xiwangbei jiushi hao $\\times$ 8 = Jiushihao Xiwangbei $\\times$ 5\", different Chinese characters represent different digits. The six-digit even number represented by \"Xiwangbei jiushi hao\" is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "256410",
"style": "rule"
} | {
"index": "7778",
"split": "train"
} |
math | [
{
"content": "Arrange the positive integers in the following number matrix:\n\\begin{tabular}{lllll}\n1 & 2 & 5 & 10 & $\\ldots$ \\\\\n4 & 3 & 6 & 11 & $\\ldots$ \\\\\n9 & 8 & 7 & 12 & $\\ldots$ \\\\\n16 & 15 & 14 & 13 & $\\ldots$ \\\\\n$\\ldots$ & $\\ldots$ & $\\ldots$ & $\\ldots$ & $\\ldots$\n\\end{tabular}\n... | {
"ground_truth": "421",
"style": "rule"
} | {
"index": "7779",
"split": "train"
} |
math | [
{
"content": "To arrange 5 volunteers and 2 elderly people in a row, where the 2 elderly people are adjacent but not at the ends, calculate the total number of different arrangements. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "960",
"style": "rule"
} | {
"index": "7780",
"split": "train"
} |
math | [
{
"content": "Please write down an irrational number whose absolute value is less than $3: \\_\\_\\_\\_\\_\\_.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "7781",
"split": "train"
} |
math | [
{
"content": "In how many distinct ways can I arrange my six keys on a keychain, if my house key must be exactly opposite my car key and my office key should be adjacent to my house key? For arrangement purposes, two placements are identical if one can be obtained from the other through rotation or flipping the... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "7782",
"split": "train"
} |
math | [
{
"content": "Calculate the following product:\n$$\\frac{1}{3}\\times9\\times\\frac{1}{27}\\times81\\times\\frac{1}{243}\\times729\\times\\frac{1}{2187}\\times6561\\times\\frac{1}{19683}\\times59049.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "243",
"style": "rule"
} | {
"index": "7783",
"split": "train"
} |
math | [
{
"content": "Each cell of a $2 \\times 5$ grid of unit squares is to be colored white or black. Compute the number of such colorings for which no $2 \\times 2$ square is a single color. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "634",
"style": "rule"
} | {
"index": "7784",
"split": "train"
} |
math | [
{
"content": "In the geometric sequence $\\{a_n\\}$, $S_4=1$, $S_8=3$, then the value of $a_{17}+a_{18}+a_{19}+a_{20}$ is. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "7785",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[ e^{2 \\pi i/17} + e^{4 \\pi i/17} + e^{6 \\pi i/17} + \\dots + e^{32 \\pi i/17}. \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "7786",
"split": "train"
} |
math | [
{
"content": "In the drawing, there is a grid consisting of 25 small equilateral triangles.\n\nHow many rhombuses can be formed from two adjacent small triangles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "7787",
"split": "train"
} |
math | [
{
"content": "Compute $63 \\times 57$ in your head. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3591",
"style": "rule"
} | {
"index": "7788",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $a\\sin B=-b\\sin \\left(A+ \\frac {\\pi}{3}\\right)$. \n$(1)$ Find $A$; \n$(2)$ If the area of $\\triangle ABC$, $S= \\frac { \\sqrt {3}}{4}c^{2}$, find the value of $\\s... | {
"ground_truth": "\\frac { \\sqrt {7}}{14}",
"style": "rule"
} | {
"index": "7789",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow {a} = (1, -3)$ and $\\overrightarrow {b} = (-2, m)$, and $\\overrightarrow {a}$ is perpendicular to ($\\overrightarrow {a} - \\overrightarrow {b}$). \n(1) Find the value of the real number $m$; \n(2) When $k\\overrightarrow {a} + \\overrightarrow {b}$ is parallel ... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "7790",
"split": "train"
} |
math | [
{
"content": "Given that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C: \\frac {x^{2}}{a^{2}} - \\frac {y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, point $P$ is on the hyperbola $C$, $PF_{2}$ is perpendicular to the x-axis, and $\\sin \\angle PF_{1}F_{2} = \\frac {1}{3}$, determine the eccentrici... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "7791",
"split": "train"
} |
math | [
{
"content": "The graph of $y=g(x)$, defined on a limited domain shown, is conceptualized through the function $g(x) = \\frac{(x-6)(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{945} - 2.5$. If each horizontal grid line represents a unit interval, determine the sum of all integers $d$ for which the equation $g(x) = d$ has exact... | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "7792",
"split": "train"
} |
math | [
{
"content": "Natural numbers \\(a, b, c\\) are chosen such that \\(a < b < c\\). It is also known that the system of equations \\(2x + y = 2035\\) and \\(y = |x - a| + |x - b| + |x - c|\\) has exactly one solution. Find the minimum possible value of \\(c\\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "1018",
"style": "rule"
} | {
"index": "7793",
"split": "train"
} |
math | [
{
"content": "If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $1000x^3+27= (ax^2 + bx +c )(d x^2 +ex + f)$ for all $x$, then what is $a^2+b^2+c^2+d^2+e^2+f^2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11,\\!090",
"style": "rule"
} | {
"index": "7794",
"split": "train"
} |
math | [
{
"content": "Compute the distance between the parallel lines given by\n\\[\\begin{pmatrix} 2 \\\\ -3 \\end{pmatrix} + t \\begin{pmatrix} 1 \\\\ -7 \\end{pmatrix}\\]and\n\\[\\begin{pmatrix} 1 \\\\ -5 \\end{pmatrix} + s \\begin{pmatrix} 1 \\\\ -7 \\end{pmatrix}.\\] Please output the final answer within \\boxed{}... | {
"ground_truth": "\\frac{9 \\sqrt{2}}{10}",
"style": "rule"
} | {
"index": "7795",
"split": "train"
} |
math | [
{
"content": "Cat and Claire are having a conversation about Cat's favorite number.\n \nCat says, \"My favorite number is a two-digit positive integer with distinct nonzero digits, $\\overline{AB}$ , such that $A$ and $B$ are both factors of $\\overline{AB}$ .\"\n \nClaire says, \"I don't know your ... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "7796",
"split": "train"
} |
math | [
{
"content": "People are standing in a circle - there are liars, who always lie, and knights, who always tell the truth. Each of them said that among the people standing next to them, there is an equal number of liars and knights. How many people are there in total if there are 48 knights? Please output the fin... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "7797",
"split": "train"
} |
math | [
{
"content": "The polynomial $f(x)$ satisfies $f(x + 1) - f(x) = 6x + 4.$ Find the leading coefficient of $f(x).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "7798",
"split": "train"
} |
math | [
{
"content": "A residential building has a construction investment of 250 yuan per square meter. Considering a service life of 50 years and an annual interest rate of 5%, calculate the monthly rent per square meter in yuan required to recoup the total investment. Please output the final answer within \\boxed{}.... | {
"ground_truth": "1.14",
"style": "rule"
} | {
"index": "7799",
"split": "train"
} |
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