data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "How many different positive three-digit integers can be formed using only the digits in the set $\\{2, 3, 5, 5, 7, 7, 7\\}$ if no digit may be used more times than it appears in the given set of available digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "43",
"style": "rule"
} | {
"index": "36000",
"split": "train"
} |
math | [
{
"content": "A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "36001",
"split": "train"
} |
math | [
{
"content": "Six students stand in a row for a photo. Among them, student A and student B are next to each other, student C is not next to either student A or student B. The number of different ways the students can stand is ______ (express the result in numbers). Please output the final answer within \\boxed{... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "36002",
"split": "train"
} |
math | [
{
"content": "Given that \\( x \\) is a real number and \\( y = \\sqrt{x^2 - 2x + 2} + \\sqrt{x^2 - 10x + 34} \\). Find the minimum value of \\( y \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{2}",
"style": "rule"
} | {
"index": "36003",
"split": "train"
} |
math | [
{
"content": "Determine the number of digits in the product of $84,123,457,789,321,005$ and $56,789,234,567,891$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "36004",
"split": "train"
} |
math | [
{
"content": "Given $a= \\int_{ 0 }^{ \\pi }(\\sin x-1+2\\cos ^{2} \\frac {x}{2})dx$, find the constant term in the expansion of $(a \\sqrt {x}- \\frac {1}{ \\sqrt {x}})^{6}\\cdot(x^{2}+2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-332",
"style": "rule"
} | {
"index": "36005",
"split": "train"
} |
math | [
{
"content": "Jack rewrites the quadratic $9x^2 - 30x - 42$ in the form of $(ax + b)^2 + c,$ where $a,$ $b,$ and $c$ are all integers. What is $ab$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-15",
"style": "rule"
} | {
"index": "36006",
"split": "train"
} |
math | [
{
"content": "Divide 6 volunteers into 4 groups, with two groups having 2 people each and the other two groups having 1 person each, to serve at four different pavilions of the World Expo. How many different allocation schemes are there? (Answer with a number). Please output the final answer within \\boxed{}.",... | {
"ground_truth": "1080",
"style": "rule"
} | {
"index": "36007",
"split": "train"
} |
math | [
{
"content": "Given an ellipse $C: \\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1 \\left( a > b > 0 \\right)$ with eccentricity $\\dfrac{\\sqrt{3}}{2}$, and the length of its minor axis is $2$.\n\n(I) Find the standard equation of the ellipse $C$.\n\n(II) Suppose a tangent line $l$ of the circle $O: x^2 + y^2 = 1$ i... | {
"ground_truth": "\\dfrac{5}{4}",
"style": "rule"
} | {
"index": "36008",
"split": "train"
} |
math | [
{
"content": "The symbol $\\diamond$ is defined so that $a \\diamond b=\\frac{a+b}{a \\times b}$. What is the value of $3 \\diamond 6$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "36009",
"split": "train"
} |
math | [
{
"content": "In acute triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\\frac{\\cos A}{\\cos C}=\\frac{a}{2b-c}$, find:<br/>\n$(1)$ The measure of angle $A$;<br/>\n$(2)$ If $a=\\sqrt{7}$, $c=3$, and $D$ is the midpoint of $BC$, find the leng... | {
"ground_truth": "\\frac{\\sqrt{19}}{2}",
"style": "rule"
} | {
"index": "36010",
"split": "train"
} |
math | [
{
"content": "Given the following conditions:①$asinB=bsin({A+\\frac{π}{3}})$; ②$S=\\frac{{\\sqrt{3}}}{2}\\overrightarrow{BA}•\\overrightarrow{CA}$; ③$c\\tan A=\\left(2b-c\\right)\\tan C$. Choose one of the three conditions and fill in the blank below, then answer the following questions.<br/>In $\\triangle ABC$... | {
"ground_truth": "\\frac{4\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "36011",
"split": "train"
} |
math | [
{
"content": "The sum of the coefficients of all terms in the expanded form of $(C_4^1x + C_4^2x^2 + C_4^3x^3 + C_4^4x^4)^2$ is 256. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "256",
"style": "rule"
} | {
"index": "36012",
"split": "train"
} |
math | [
{
"content": "Find all pairs $(x,y)$ of nonnegative integers that satisfy \\[x^3y+x+y=xy+2xy^2.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(0, 0), (1, 1), (2, 2)",
"style": "rule"
} | {
"index": "36013",
"split": "train"
} |
math | [
{
"content": "How many prime numbers have squares between 4000 and 7000? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "36014",
"split": "train"
} |
math | [
{
"content": "Under normal circumstances, for people aged between 18 and 38, the regression equation of weight $y$ (kg) to height $x$ (cm) is $\\overset{\\land }{y} = 0.72x - 58.2$. Zhang Hong, who is 20 years old and has a height of 178 cm, should have a weight of approximately \\_\\_\\_\\_\\_ kg. Please outpu... | {
"ground_truth": "69.96",
"style": "rule"
} | {
"index": "36015",
"split": "train"
} |
math | [
{
"content": "Let $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a s_k + b s_{k-1} + c s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$? Please output the final an... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "36016",
"split": "train"
} |
math | [
{
"content": "A waiter at the restaurant U Šejdíře always adds the current date to the bill: he increases the total amount spent by as many crowns as the day of the month it is.\n\nIn September, a group of three friends dined at the restaurant twice. The first time, each person paid separately, and the waiter a... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "36017",
"split": "train"
} |
math | [
{
"content": "Given the complex number $z= \\frac {(1+i)^{2}+2(5-i)}{3+i}$.\n$(1)$ Find $|z|$;\n$(2)$ If $z(z+a)=b+i$, find the values of the real numbers $a$ and $b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-13",
"style": "rule"
} | {
"index": "36018",
"split": "train"
} |
math | [
{
"content": "For each integer $n\\ge 2$ , determine, with proof, which of the two positive real numbers $a$ and $b$ satisfying \\[a^n=a+1,\\qquad b^{2n}=b+3a\\] is larger. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\[ a > b \\]",
"style": "rule"
} | {
"index": "36019",
"split": "train"
} |
math | [
{
"content": "If $\\left(k-1\\right)x^{|k|}+3\\geqslant 0$ is a one-variable linear inequality about $x$, then the value of $k$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "36020",
"split": "train"
} |
math | [
{
"content": "A box contains three balls, each of a different color. Every minute, Randall randomly draws a ball from the box, notes its color, and then returns it to the box. Consider the following two conditions: (1) Some ball has been drawn at least three times (not necessarily consecutively). (2) Every ball... | {
"ground_truth": "\\frac{13}{27}",
"style": "rule"
} | {
"index": "36021",
"split": "train"
} |
math | [
{
"content": "There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \\text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each ... | {
"ground_truth": "1024",
"style": "rule"
} | {
"index": "36022",
"split": "train"
} |
math | [
{
"content": "Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are $6$ cm in diameter and $12$ cm high. Felicia buys cat food in cylindrical cans that are $12$ cm in diameter and $6$ cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felici... | {
"ground_truth": "1:2",
"style": "rule"
} | {
"index": "36023",
"split": "train"
} |
math | [
{
"content": "Given two lines $l_{1}$: $(m+1)x+2y+2m-2=0$ and $l_{2}$: $2x+(m-2)y+2=0$, if $l_{1} \\parallel l_{2}$, then $m=$ ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "36024",
"split": "train"
} |
math | [
{
"content": "On a table, there are 20 cards numbered from 1 to 20. Each time, Xiao Ming picks out 2 cards such that the number on one card is 2 more than twice the number on the other card. What is the maximum number of cards Xiao Ming can pick? Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "36025",
"split": "train"
} |
math | [
{
"content": "A 60-degree angle contains five circles, where each subsequent circle (starting from the second) touches the previous one. By how many times is the sum of the areas of all five circles greater than the area of the smallest circle? Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "7381",
"style": "rule"
} | {
"index": "36026",
"split": "train"
} |
math | [
{
"content": "Mr. Pipkins said, \"I was walking along the road at a speed of $3 \\frac{1}{2}$ km/h when suddenly a car sped past me, almost knocking me off my feet.\"\n\n\"What was its speed?\" his friend asked.\n\n\"I can tell you now. From the moment it sped past me until it disappeared around the bend, I too... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "36027",
"split": "train"
} |
math | [
{
"content": "Kanga labelled the vertices of a square-based pyramid using \\(1, 2, 3, 4,\\) and \\(5\\) once each. For each face, Kanga calculated the sum of the numbers on its vertices. Four of these sums equaled \\(7, 8, 9,\\) and \\(10\\). What is the sum for the fifth face?\nA) 11\nB) 12\nC) 13\nD) 14\nE) 1... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "36028",
"split": "train"
} |
math | [
{
"content": "A weight of 200 grams rests on a table. It was flipped and placed on the table on its other side, the area of which is 15 sq. cm smaller. As a result, the pressure on the table increased by 1200 Pa. Find the area of the side on which the weight originally rested. Give your answer in sq. cm, rounde... | {
"ground_truth": "24.85",
"style": "rule"
} | {
"index": "36029",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=2 \\sqrt {3}\\sin x\\cos x-\\cos 2x$, where $x\\in R$.\n(1) Find the interval where the function $f(x)$ is monotonically increasing.\n(2) In $\\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $f(A)=2$, $C=... | {
"ground_truth": "\\frac {3+ \\sqrt {3}}{2}",
"style": "rule"
} | {
"index": "36030",
"split": "train"
} |
math | [
{
"content": "Four cars $A, B, C,$ and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ travel clockwise, while $C$ and $D$ travel counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meets $C$ for the first ... | {
"ground_truth": "53",
"style": "rule"
} | {
"index": "36031",
"split": "train"
} |
math | [
{
"content": "A club has 12 members, and wishes to pick a president, a vice-president, a secretary, and a treasurer. However, the president and vice-president must have been members of the club for at least 3 years. If 4 of the existing members meet this criterion, in how many ways can these positions be filled... | {
"ground_truth": "1080",
"style": "rule"
} | {
"index": "36032",
"split": "train"
} |
math | [
{
"content": "There are five unmarked envelopes on a table, each with a letter for a different person. If the mail is randomly distributed to these five people, with each person getting one letter, what is the probability that exactly four people get the right letter? Please output the final answer within \\bo... | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "36033",
"split": "train"
} |
math | [
{
"content": "What is the least integer whose square is 12 more than three times the number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "36034",
"split": "train"
} |
math | [
{
"content": "Suppose $a$ and $b$ are positive integers such that $\\gcd(a,b)$ is divisible by exactly $7$ distinct primes and $\\mathop{\\text{lcm}}[a,b]$ is divisible by exactly $28$ distinct primes.\n\nIf $a$ has fewer distinct prime factors than $b$, then $a$ has at most how many distinct prime factors? Ple... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "36035",
"split": "train"
} |
math | [
{
"content": "What is the greatest product obtainable from two integers whose sum is 2016? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1016064",
"style": "rule"
} | {
"index": "36036",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $a \\neq b$, $c= \\sqrt{3}$, and $\\cos^2A - \\cos^2B = \\sqrt{3}\\sin A\\cos A - \\sqrt{3}\\sin B\\cos B$.\n\n$(I)$ Find the magnitude of angle $C$.\n\n$(II)$ If $\\... | {
"ground_truth": "\\frac{8\\sqrt{3}+18}{25}",
"style": "rule"
} | {
"index": "36037",
"split": "train"
} |
math | [
{
"content": "Let $S_{n}$ and $T_{n}$ represent the sum of the first $n$ terms of the arithmetic sequences ${a_{n}}$ and ${b_{n}}$, respectively. Given that $\\frac{S_{n}}{T_{n}} = \\frac{n+1}{2n-1}$, where $n \\in \\mathbb{N}^*$, find the value of $\\frac{a_{3} + a_{7}}{b_{1} + b_{9}}$. Please output the final... | {
"ground_truth": "\\frac{10}{17}",
"style": "rule"
} | {
"index": "36038",
"split": "train"
} |
math | [
{
"content": "In a table tennis match between player A and player B, the \"best of five sets\" rule is applied, which means the first player to win three sets wins the match. If the probability of player A winning a set is $\\dfrac{2}{3}$, and the probability of player B winning a set is $\\dfrac{1}{3}$, then t... | {
"ground_truth": "\\dfrac{8}{27}",
"style": "rule"
} | {
"index": "36039",
"split": "train"
} |
math | [
{
"content": "The first four terms in an arithmetic sequence are $x + y, x - y, xy,$ and $x/y,$ in that order. What is the fifth term? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{123}{40}",
"style": "rule"
} | {
"index": "36040",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, if $a=3$, $b= \\sqrt {3}$, $\\angle A= \\dfrac {\\pi}{3}$, then the size of $\\angle C$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac {\\pi}{2}",
"style": "rule"
} | {
"index": "36041",
"split": "train"
} |
math | [
{
"content": "What is the largest integer $n$ for which $\\binom{8}{3} + \\binom{8}{4} = \\binom{9}{n}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "36042",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\cos (2x-\\frac{\\pi }{3})+2\\sin^2x$.\n\n(Ⅰ) Find the period of the function $f(x)$ and the intervals where it is monotonically increasing;\n\n(Ⅱ) When $x \\in [0,\\frac{\\pi}{2}]$, find the maximum and minimum values of the function $f(x)$. Please output the final answe... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "36043",
"split": "train"
} |
math | [
{
"content": "A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\\log_{2}x, \\log_{3}x,$ and $\\log_{4}x.$ What is $x?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "576",
"style": "rule"
} | {
"index": "36044",
"split": "train"
} |
math | [
{
"content": "Given the following arrays, each composed of three numbers: $(1,2,3)$, $(2,4,6)$, $(3,8,11)$, $(4,16,20)$, $(5,32,37)$, ..., $(a\\_n,b\\_n,c\\_n)$. If the sum of the first $n$ terms of the sequence $\\{c\\_n\\}$ is denoted as $M\\_n$, find the value of $M\\_{10}$. Please output the final answer wi... | {
"ground_truth": "2101",
"style": "rule"
} | {
"index": "36045",
"split": "train"
} |
math | [
{
"content": "Pizzas are sized by diameter. What percent increase in area results if Chantel's pizza increases from a 10-inch pizza to a 12-inch pizza? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "44\\%",
"style": "rule"
} | {
"index": "36046",
"split": "train"
} |
math | [
{
"content": "A line parallel to the base of a triangle divides it into parts whose areas are in the ratio $2:1$, counting from the vertex. In what ratio does this line divide the sides of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(\\sqrt{6} + 2) : 1",
"style": "rule"
} | {
"index": "36047",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of $9^x - 3^x + 1$ over all real numbers $x.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "36048",
"split": "train"
} |
math | [
{
"content": "If $m+1=\frac{n-2}{3}$, what is the value of $3 m-n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "36049",
"split": "train"
} |
math | [
{
"content": "In right triangle $ABC$ with $\\angle A = 90^\\circ$, we have $AB =16$ and $BC = 24$. Find $\\sin A$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "36050",
"split": "train"
} |
math | [
{
"content": "Suppose that \\(p\\) and \\(q\\) are two different prime numbers and that \\(n=p^{2} q^{2}\\). What is the number of possible values of \\(n\\) with \\(n<1000\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "36051",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer with exactly $12$ positive factors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "36052",
"split": "train"
} |
math | [
{
"content": "In the diagram, if $\\angle PQR = 48^\\circ$, what is the measure of $\\angle PMN$? [asy]\n\nsize(6cm);\n\npair p = (0, 0); pair m = dir(180 - 24); pair n = dir(180 + 24); pair r = 1.3 * dir(24); pair q = 2 * 1.3 * Cos(48) * dir(-24);\n\nlabel(\"$M$\", m, N); label(\"$R$\", r, N); label(\"$P$\", p... | {
"ground_truth": "66^\\circ",
"style": "rule"
} | {
"index": "36053",
"split": "train"
} |
math | [
{
"content": "The water tank in the diagram below is in the shape of an inverted right circular cone. The radius of its base is 8 feet, and its height is 64 feet. The water in the tank is $40\\%$ of the tank's capacity. The height of the water in the tank can be written in the form $a\\sqrt[3]{b}$, where $a$ an... | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "36054",
"split": "train"
} |
math | [
{
"content": "The exterior angles of a triangle are proportional to the numbers 5: 7: 8. Find the angle between the altitudes of this triangle drawn from the vertices of its smaller angles. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "36055",
"split": "train"
} |
math | [
{
"content": "A cuckoo clock chimes the number of times corresponding to the current hour (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it was 9:05 and started moving the minute hand forward until the clock read 7 hours later. How many times did the cuckoo chime during this p... | {
"ground_truth": "43",
"style": "rule"
} | {
"index": "36056",
"split": "train"
} |
math | [
{
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\\times 5$ square array of dots? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "36057",
"split": "train"
} |
math | [
{
"content": "Compute $\\sin 12^\\circ \\sin 36^\\circ \\sin 72^\\circ \\sin 84^\\circ.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{16}",
"style": "rule"
} | {
"index": "36058",
"split": "train"
} |
math | [
{
"content": "Given $\\sin x = \\frac{3}{5}$, with $x \\in \\left( \\frac{\\pi}{2}, \\pi \\right)$, find the values of $\\cos 2x$ and $\\tan\\left( x + \\frac{\\pi}{4} \\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{7}",
"style": "rule"
} | {
"index": "36059",
"split": "train"
} |
math | [
{
"content": "Rationalize the denominator: $\\frac{1}{\\sqrt{2}-1}$. Express your answer in simplest form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}+1",
"style": "rule"
} | {
"index": "36060",
"split": "train"
} |
math | [
{
"content": "Let $n$ be a positive integer such that $12n^2+12n+11$ is a $4$ -digit number with all $4$ digits equal. Determine the value of $n$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "36061",
"split": "train"
} |
math | [
{
"content": "A sequence of twelve \\(0\\)s and/or \\(1\\)s is randomly generated and must start with a '1'. If the probability that this sequence does not contain two consecutive \\(1\\)s can be written in the form \\(\\dfrac{m}{n}\\), where \\(m,n\\) are relatively prime positive integers, find \\(m+n\\). Ple... | {
"ground_truth": "2281",
"style": "rule"
} | {
"index": "36062",
"split": "train"
} |
math | [
{
"content": "What is the sum of the tens digit and the ones digit of the integer form of $(2+3)^{23}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "36063",
"split": "train"
} |
math | [
{
"content": "The measure of angle $ACB$ is 60 degrees. If ray $CA$ is rotated 630 degrees about point $C$ in a counterclockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "36064",
"split": "train"
} |
math | [
{
"content": "The administrator accidentally mixed up the keys for 10 rooms. If each key can only open one room, what is the maximum number of attempts needed to match all keys to their corresponding rooms? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "36065",
"split": "train"
} |
math | [
{
"content": "Mr. Fast eats a pizza slice in 5 minutes, Mr. Slow eats a pizza slice in 10 minutes, and Mr. Steady eats a pizza slice in 8 minutes. How long does it take for them to eat 24 pizza slices together? Express your answer in minutes. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "36066",
"split": "train"
} |
math | [
{
"content": "Let $x^2+bx+c = 0$ be a quadratic whose roots are each two more than the roots of $3x^2-5x-7$. What is $c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "36067",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC, AB = 13, BC = 14$ and $CA = 15$. Also, $M$ is the midpoint of side $AB$ and $H$ is the foot of the altitude from $A$ to $BC$. \nThe length of $HM$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6.5",
"style": "rule"
} | {
"index": "36068",
"split": "train"
} |
math | [
{
"content": "(1) Simplify: $\\dfrac{\\tan(3\\pi-\\alpha)\\cos(2\\pi-\\alpha)\\sin(-\\alpha+\\dfrac{3\\pi}{2})}{\\cos(-\\alpha-\\pi)\\sin(-\\pi+\\alpha)\\cos(\\alpha+\\dfrac{5\\pi}{2})}$;\n\n(2) Given $\\tan\\alpha=\\dfrac{1}{4}$, find the value of $\\dfrac{1}{2\\cos^{2}\\alpha-3\\sin\\alpha\\cos\\alpha}$. Plea... | {
"ground_truth": "\\dfrac{17}{20}",
"style": "rule"
} | {
"index": "36069",
"split": "train"
} |
math | [
{
"content": "Let $x_1, x_2, \\ldots , x_n$ be a sequence of integers such that\n(i) $-1 \\le x_i \\le 2$ for $i = 1,2, \\ldots n$\n(ii) $x_1 + \\cdots + x_n = 19$; and\n(iii) $x_1^2 + x_2^2 + \\cdots + x_n^2 = 99$.\nLet $m$ and $M$ be the minimal and maximal possible values of $x_1^3 + \\cdots + x_n^3$, respec... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "36070",
"split": "train"
} |
math | [
{
"content": "Express the number one million using numbers that contain only the digit 9 and the algebraic operations of addition, subtraction, multiplication, division, powers, and roots. Determine at least three different solutions. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1000000",
"style": "rule"
} | {
"index": "36071",
"split": "train"
} |
math | [
{
"content": "Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ be real numbers whose sum is 20. Determine with proof the smallest possible value of \\(\\sum_{1 \\leq i<j \\leq 5}\\left\\lfloor a_{i}+a_{j}\\right\\rfloor\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\[\n72\n\\]",
"style": "rule"
} | {
"index": "36072",
"split": "train"
} |
math | [
{
"content": "Find the total perimeter of the combined shape of the flower bed, given that the garden is shaped as a right triangle adjacent to a rectangle, the hypotenuse of the triangle coincides with one of the sides of the rectangle, the triangle has legs of lengths 3 meters and 4 meters, and the rectangle ... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "36073",
"split": "train"
} |
math | [
{
"content": "We wish to color the integers $1,2,3, \\ldots, 10$ in red, green, and blue, so that no two numbers $a$ and $b$, with $a-b$ odd, have the same color. (We do not require that all three colors be used.) In how many ways can this be done? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "186",
"style": "rule"
} | {
"index": "36074",
"split": "train"
} |
math | [
{
"content": "A cao has 6 legs, 3 on each side. A walking pattern for the cao is defined as an ordered sequence of raising and lowering each of the legs exactly once (altogether 12 actions), starting and ending with all legs on the ground. The pattern is safe if at any point, he has at least 3 legs on the groun... | {
"ground_truth": "1416528",
"style": "rule"
} | {
"index": "36075",
"split": "train"
} |
math | [
{
"content": "Define the sequence $a_{1}, a_{2} \\ldots$ as follows: $a_{1}=1$ and for every $n \\geq 2$, $a_{n}= \\begin{cases}n-2 & \\text { if } a_{n-1}=0 \\\\ a_{n-1}-1 & \\text { if } a_{n-1} \\neq 0\\end{cases}$. A non-negative integer $d$ is said to be jet-lagged if there are non-negative integers $r, s$... | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "36076",
"split": "train"
} |
math | [
{
"content": "What is the units digit of the sum of the squares of the first 2007 odd, positive integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "36077",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6, BC=5=DA,$and $CD=4.$ Draw circles of radius 3 centered at $A$ and $B,$ and circles of radius 2 centered at $C$ and $D.$ A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\\frac{-k+m\\s... | {
"ground_truth": "134",
"style": "rule"
} | {
"index": "36078",
"split": "train"
} |
math | [
{
"content": "West, Non-West, Russia:\n1st place - Russia: 302790.13 cubic meters/person\n2nd place - Non-West: 26848.55 cubic meters/person\n3rd place - West: 21428 cubic meters/person Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "302790.13",
"style": "rule"
} | {
"index": "36079",
"split": "train"
} |
math | [
{
"content": "Two distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 5\\}$. What is the probability that their product is an even number? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{10}",
"style": "rule"
} | {
"index": "36080",
"split": "train"
} |
math | [
{
"content": "An employer hired a worker for a year and promised to give him 12 rubles and a coat. However, the worker wanted to leave after 7 months. Upon settlement, he received the coat and 5 rubles. How much did the coat cost? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.8",
"style": "rule"
} | {
"index": "36081",
"split": "train"
} |
math | [
{
"content": "The famous Italian mathematician Fibonacci, while studying the problem of rabbit population growth, discovered a sequence of numbers: 1, 1, 2, 3, 5, 8, 13, ..., where starting from the third number, each number is equal to the sum of the two numbers preceding it. This sequence of numbers $\\{a_n\\... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "36082",
"split": "train"
} |
math | [
{
"content": "Given that the domain of $f(x)$ is $\\mathbb{R}$, $f(1)=\\frac{1}{4}$, and it satisfies the equation $4f(x)f(y) = f(x+y) + f(x-y)$, find the value of $f(2016)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "36083",
"split": "train"
} |
math | [
{
"content": "In the polar coordinate system $Ox$, the polar equation of curve $C_{1}$ is $ρ=\\frac{2\\sqrt{2}}{sin(θ+\\frac{π}{4})}$, with the pole $O$ as the origin and the polar axis $Ox$ as the x-axis. A Cartesian coordinate system $xOy$ is established with the same unit length. It is known that the general... | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "36084",
"split": "train"
} |
math | [
{
"content": "Let $x=-2016$. What is the value of $| ||x|-x|-|x| | -x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4032",
"style": "rule"
} | {
"index": "36085",
"split": "train"
} |
math | [
{
"content": "Below is pictured a regular seven-pointed star. Find the measure of angle \\( a \\) in radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\pi}{7}",
"style": "rule"
} | {
"index": "36086",
"split": "train"
} |
math | [
{
"content": "A right-angled triangle has sides of lengths 6, 8, and 10. A circle is drawn so that the area inside the circle but outside this triangle equals the area inside the triangle but outside the circle. The radius of the circle is closest to: Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "2.8",
"style": "rule"
} | {
"index": "36087",
"split": "train"
} |
math | [
{
"content": "We have a cube with 4 blue faces and 2 red faces. What's the probability that when it is rolled, a blue face will be facing up? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "36088",
"split": "train"
} |
math | [
{
"content": "What percentage error do we make if we approximate the side of a regular heptagon by taking half of the chord corresponding to the $120^\\circ$ central angle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.2",
"style": "rule"
} | {
"index": "36089",
"split": "train"
} |
math | [
{
"content": "Two positive rational numbers $x$ and $y$, when written in lowest terms, have the property that the sum of their numerators is 9 and the sum of their denominators is 10 . What is the largest possible value of $x+y$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "73 / 9",
"style": "rule"
} | {
"index": "36090",
"split": "train"
} |
math | [
{
"content": "In the tetrahedron $P-ABC$, edges $PA$, $AB$, and $AC$ are mutually perpendicular, and $PA = AB = AC$. Points $E$ and $F$ are the midpoints of segments $AB$ and $PC$, respectively. Find the sine of the angle between line $EF$ and plane $PBC$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "36091",
"split": "train"
} |
math | [
{
"content": "Given that the positive real numbers \\( u, v, \\) and \\( w \\) are all not equal to 1, if \\(\\log _{u} (v w)+\\log _{v} w=5\\) and \\(\\log _{v} u+\\log _{w} v=3\\), then find the value of \\(\\log _{w} u\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4/5",
"style": "rule"
} | {
"index": "36092",
"split": "train"
} |
math | [
{
"content": "Given plane vectors $\\overrightarrow{a}=(-1,2)$ and $\\overrightarrow{b}=(1,-4)$. \n$(1)$ If $4\\overrightarrow{a}+\\overrightarrow{b}$ is perpendicular to $k\\overrightarrow{a}-\\overrightarrow{b}$, find the value of the real number $k$. \n$(2)$ If $\\theta$ is the angle between $4\\overrighta... | {
"ground_truth": "-\\frac{3}{4}",
"style": "rule"
} | {
"index": "36093",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ is an equilateral triangle with each side of length 9. Points $D$, $E$, and $F$ are the trisection points of sides $AB$, $BC$, and $CA$ respectively, such that $D$ is closer to $A$, $E$ is closer to $B$, and $F$ is closer to $C$. Point $G$ is the midpoint of segment $DF$ and point $... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "36094",
"split": "train"
} |
math | [
{
"content": "An equilateral triangle has an area of $64\\sqrt{3}$ $\\text{cm}^2$. If each side of the triangle is decreased by 4 cm, by how many square centimeters is the area decreased? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28\\sqrt{3}",
"style": "rule"
} | {
"index": "36095",
"split": "train"
} |
math | [
{
"content": "Convert the point $(\\rho,\\theta,\\phi) = \\left( 4, \\frac{5 \\pi}{3}, \\frac{\\pi}{2} \\right)$ in spherical coordinates to rectangular coordinates. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(2, -2 \\sqrt{3}, 0)",
"style": "rule"
} | {
"index": "36096",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to $\\angle A$, $\\angle B$, and $\\angle C$ are $a$, $b$, and $c$ respectively. $M$ is the midpoint of $BC$ with $BM=MC=2$, and $AM=b-c$. Find the maximum area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "36097",
"split": "train"
} |
math | [
{
"content": "Compute the number of real solutions $(x, y, z, w)$ to the system of equations: $$\\begin{array}{rlrl} x & =z+w+z w x & z & =x+y+x y z \\\\ y & =w+x+w x y & w & =y+z+y z w \\end{array}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "36098",
"split": "train"
} |
math | [
{
"content": "Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Then $n$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "36099",
"split": "train"
} |
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