data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "What is the sum of the tens digit and the ones digit of the integer form of $(2+4)^{15}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "3800",
"split": "train"
} |
math | [
{
"content": "Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible? Pleas... | {
"ground_truth": "132",
"style": "rule"
} | {
"index": "3801",
"split": "train"
} |
math | [
{
"content": "Cube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\\overline{AB}$ and $\\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written i... | {
"ground_truth": "89",
"style": "rule"
} | {
"index": "3802",
"split": "train"
} |
math | [
{
"content": "There are 300 children in the \"Young Photographer\" club. In a session, they divided into 100 groups of 3 people each, and in every group, each member took a photograph of the other two members in their group. No one took any additional photographs. In total, there were 100 photographs of \"boy+b... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "3803",
"split": "train"
} |
math | [
{
"content": "Given $A(a, 1-a^{2})$, $B(b, 1-b^{2})$, where $ab \\lt 0$, tangents are drawn at points $A$ and $B$ to the parabola $y=1-x^{2}$. The minimum area enclosed by the two tangents and the $x$-axis is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8\\sqrt{3}}{9}",
"style": "rule"
} | {
"index": "3804",
"split": "train"
} |
math | [
{
"content": "Given $|\\vec{a}|=1$, $|\\vec{b}|=2$, and the angle between $\\vec{a}$ and $\\vec{b}$ is $60^{\\circ}$.\n$(1)$ Find $\\vec{a}\\cdot \\vec{b}$, $(\\vec{a}- \\vec{b})\\cdot(\\vec{a}+ \\vec{b})$;\n$(2)$ Find $|\\vec{a}- \\vec{b}|$. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "3805",
"split": "train"
} |
math | [
{
"content": "A king traversed a $9 \\times 9$ chessboard, visiting each square exactly once. The king's route is not a closed loop and may intersect itself. What is the maximum possible length of such a route if the length of a move diagonally is $\\sqrt{2}$ and the length of a move vertically or horizontally ... | {
"ground_truth": "16 + 64 \\sqrt{2}",
"style": "rule"
} | {
"index": "3806",
"split": "train"
} |
math | [
{
"content": "The sides of a triangle have lengths \\( 13, 17, \\) and \\( k, \\) where \\( k \\) is a positive integer. For how many values of \\( k \\) is the triangle obtuse? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "3807",
"split": "train"
} |
math | [
{
"content": "Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal of the field and saved a distance equal to $\\frac{1}{2}$ the longer side. The ratio of the shorter side of the rectangle to the longer side was: Please output the final answer within \\box... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "3808",
"split": "train"
} |
math | [
{
"content": "A certain department store sells a batch of shirts. The cost price of each shirt is $80. On average, 30 shirts can be sold per day, with a profit of $50 per shirt. In order to increase sales and profits, the store decides to take appropriate price reduction measures. After investigation, it is fou... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "3809",
"split": "train"
} |
math | [
{
"content": "A domino is a rectangular tile composed of two squares. An integer is represented on both squares, and each integer 0-9 is paired with every integer 0-9 exactly once to form a complete set. A $\\textit{double}$ is a domino that has the same integer on both of its squares. What is the probability t... | {
"ground_truth": "\\frac{2}{11}",
"style": "rule"
} | {
"index": "3810",
"split": "train"
} |
math | [
{
"content": "Given that the random variable $\\xi$ follows a normal distribution $N(0,\\sigma^{2})$, and $P(\\xi > 2)=0.023$, determine $P(-2\\leqslant \\xi\\leqslant 2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.954",
"style": "rule"
} | {
"index": "3811",
"split": "train"
} |
math | [
{
"content": "When a student used a calculator to find the average of 30 data points, they mistakenly entered one of the data points, 105, as 15. Find the difference between the calculated average and the actual average. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "3812",
"split": "train"
} |
math | [
{
"content": "Let $p$, $q$, $r$, $s$, and $t$ be positive integers such that $p+q+r+s+t=2022$. Let $N$ be the largest of the sum $p+q$, $q+r$, $r+s$, and $s+t$. What is the smallest possible value of $N$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "506",
"style": "rule"
} | {
"index": "3813",
"split": "train"
} |
math | [
{
"content": "Let the operation $\\#$ be defined as $\\#(a, b, c) = b^2 - 4ac$, for all real numbers $a, b$, and $c$. Define a new operation $\\oplus$ by $\\oplus(a, b, c, d) = \\#(a, b + d, c) - \\#(a, b, c)$. What is the value of $\\oplus(2, 4, 1, 3)$? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "3814",
"split": "train"
} |
math | [
{
"content": "A truck travels $\\frac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10b}{t}",
"style": "rule"
} | {
"index": "3815",
"split": "train"
} |
math | [
{
"content": "Given $a \\star b = a^2 + 2ab + b^2$, what is the value of $a \\star b$ when $a = 4$ and $b = 6?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "3816",
"split": "train"
} |
math | [
{
"content": "Four boxes with ball capacity 3, 5, 7, and 8 are given. Find the number of ways to distribute 19 identical balls into these boxes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "3817",
"split": "train"
} |
math | [
{
"content": "Given a triangle $\\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, and $\\frac{a}{b}=\\frac{1+\\cos A}{\\cos C}$.\n(1) Find angle $A$;\n(2) If $a=1$, find the maximum area $S$ of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "3818",
"split": "train"
} |
math | [
{
"content": "How many positive integers between 10 and 2016 are divisible by 3 and have all of their digits the same? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "3819",
"split": "train"
} |
math | [
{
"content": "Insert a square into an isosceles triangle with a lateral side of 10 and a base of 12. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.8",
"style": "rule"
} | {
"index": "3820",
"split": "train"
} |
math | [
{
"content": "Eleven is the fifth prime number. What is the tenth prime number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "3821",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, angle $A$ is $90^\\circ$, $BC = 10$ and $\\tan C = 3\\cos B$. What is $AB$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{20\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "3822",
"split": "train"
} |
math | [
{
"content": "Given $\\sin (α- \\frac{π}{3})= \\frac{3}{5} $, where $a∈(\\frac{π}{4}, \\frac{π}{2})$, find $\\tan α=\\_\\_\\_\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "- \\frac{48+25 \\sqrt{3}}{11}",
"style": "rule"
} | {
"index": "3823",
"split": "train"
} |
math | [
{
"content": "In the movie \"The Wandering Earth 2,\" there are many UEG (United Earth Government) mechanical devices that are drivable, operable, and deformable, all of which are from the leading Chinese engineering machinery brand - XCMG. Many of the hardcore equipment in the movie are not special effects, bu... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "3824",
"split": "train"
} |
math | [
{
"content": "Earl and Bob start their new jobs on the same day. Earl's work schedule is to work for 3 days followed by 1 day off, while Bob's work schedule is to work for 7 days followed by 3 days off. In the first 1000 days, how many days off do they have in common? Please output the final answer within \\box... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "3825",
"split": "train"
} |
math | [
{
"content": "Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\[\n\\boxed{2, 3, 7}\n\\]",
"style": "rule"
} | {
"index": "3826",
"split": "train"
} |
math | [
{
"content": "The Tigers beat the Sharks 2 out of the 3 times they played. They then played $N$ more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for $N$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "3827",
"split": "train"
} |
math | [
{
"content": "Two couples each bring one child to visit the zoo. After purchasing tickets, they line up to enter the zoo one by one. For safety reasons, the two fathers must be positioned at the beginning and the end of the line. Moreover, the two children must be positioned together. Determine the number of di... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "3828",
"split": "train"
} |
math | [
{
"content": "The measures of the interior angles of a convex hexagon form an increasing arithmetic sequence. How many such sequences are possible if the hexagon is not equiangular and all of the angle degree measures are positive integers less than $150$ degrees? Please output the final answer within \\boxed{}... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "3829",
"split": "train"
} |
math | [
{
"content": "From the natural numbers 1 to 100, each time we take out two different numbers so that their sum is greater than 100, how many different ways are there to do this? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2500",
"style": "rule"
} | {
"index": "3830",
"split": "train"
} |
math | [
{
"content": "Given that 800 students were surveyed, and their pasta and pizza preferences were lasagna (150 students), manicotti (120 students), ravioli (180 students), spaghetti (200 students), and pizza (150 students), calculate the ratio of the number of students who preferred spaghetti to the number of stu... | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "3831",
"split": "train"
} |
math | [
{
"content": "Two circles with centers $A$ and $B$ and radii 2 and 1, respectively, are tangent to each other. Point $C$ lies on a line that is tangent to each of the circles and is at a distance of $\\frac{3 \\sqrt{3}}{2 \\sqrt{2}}$ from the midpoint of segment $AB$. Find the area $S$ of triangle $ABC$, given ... | {
"ground_truth": "\\frac{15 \\sqrt{2}}{8}",
"style": "rule"
} | {
"index": "3832",
"split": "train"
} |
math | [
{
"content": "In our province, the new college entrance examination adopts a \"choose 3 out of 7\" model, which means choosing 3 subjects from politics, history, geography, physics, chemistry, biology, and technology as elective subjects. How many possible combinations of elective subjects are there? If person ... | {
"ground_truth": "92",
"style": "rule"
} | {
"index": "3833",
"split": "train"
} |
math | [
{
"content": "The number \\( N \\) has the smallest positive divisor 1, the second largest positive divisor \\( k \\), and the third largest positive divisor \\( m \\). Moreover, \\( k^k + m^m = N \\). What is \\( N \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "260",
"style": "rule"
} | {
"index": "3834",
"split": "train"
} |
math | [
{
"content": "A certain agency has 18 elderly, 12 middle-aged, and 6 young individuals. When drawing a sample of size X using systematic sampling and stratified sampling, there is no need to discard any individuals. However, if the sample size is increased by 1, then using systematic sampling requires the remov... | {
"ground_truth": "X = 6",
"style": "rule"
} | {
"index": "3835",
"split": "train"
} |
math | [
{
"content": "If $e^{i \\theta} = \\frac{3 + i \\sqrt{8}}{4},$ then find $\\sin 6 \\theta.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{855 \\sqrt{2}}{1024}",
"style": "rule"
} | {
"index": "3836",
"split": "train"
} |
math | [
{
"content": "Let \\( a \\) and \\( b \\) be the roots of \\( x^2 + 2000x + 1 = 0 \\) and let \\( c \\) and \\( d \\) be the roots of \\( x^2 - 2008x + 1 = 0 \\). Find the value of \\( (a+c)(b+c)(a-d)(b-d) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32064",
"style": "rule"
} | {
"index": "3837",
"split": "train"
} |
math | [
{
"content": "Given a complex number $z$ satisfying $z+ \\bar{z}=6$ and $|z|=5$.\n\n$(1)$ Find the imaginary part of the complex number $z$;\n\n$(2)$ Find the real part of the complex number $\\dfrac{z}{1-i}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{7}{2}",
"style": "rule"
} | {
"index": "3838",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, it is known that $\\sin A : \\sin B : \\sin C = 3 : 5 : 7$. The largest interior angle of this triangle is equal to ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\pi}{3}",
"style": "rule"
} | {
"index": "3839",
"split": "train"
} |
math | [
{
"content": "A square grid $100 \\times 100$ is tiled in two ways - only with dominoes and only with squares $2 \\times 2$. What is the least number of dominoes that are entirely inside some square $2 \\times 2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "3840",
"split": "train"
} |
math | [
{
"content": "For a nonnegative integer $n$, let $r_7(n)$ denote the remainder when $n$ is divided by $7.$ Determine the $15^{\\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\\le 3.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "3841",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than 2023 are congruent to 7 modulo 13? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "156",
"style": "rule"
} | {
"index": "3842",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $AB = 8$, $BC = 8$, and $CA = 6$.\n\nPoint $P$ is randomly selected inside triangle $ABC$. What is the probability that $P$ is closer to vertex $C$ than it is to either vertex $A$ or $B$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "3843",
"split": "train"
} |
math | [
{
"content": "João, Jorge, José, and Jânio are good friends. One time, João was out of money, but his friends had some. So Jorge gave João a fifth of his money, José gave João a fourth of his money, and Jânio gave João a third of his money. If all of them gave the same amount of money to João, what fraction of ... | {
"ground_truth": "1/4",
"style": "rule"
} | {
"index": "3844",
"split": "train"
} |
math | [
{
"content": "A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately? Please output the final answer within... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "3845",
"split": "train"
} |
math | [
{
"content": "In triangle $A.\\sqrt{21}$, $2\\sin 2A\\cos A-\\sin 3A+\\sqrt{3}\\cos A=\\sqrt{3}$\n\n$(1)$ Find the size of angle $A$;\n\n$(2)$ Given that $a,b,c$ are the sides opposite to angles $A,B,C$ respectively, and $a=1$ and $\\sin A+\\sin (B-C)=2\\sin 2C$, find the area of triangle $A.\\sqrt{21}$. Please... | {
"ground_truth": "\\frac{\\sqrt{3}}{6}",
"style": "rule"
} | {
"index": "3846",
"split": "train"
} |
math | [
{
"content": "The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x \\neq y$ is the harmonic mean of $x$ and $y$ equal to $4^{15}$? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "3847",
"split": "train"
} |
math | [
{
"content": "A bag contains 15 red marbles, 10 blue marbles, and 5 green marbles. Four marbles are selected at random and without replacement. What is the probability that two marbles are red, one is blue, and one is green? Express your answer as a common fraction. Please output the final answer within \\boxed... | {
"ground_truth": "\\frac{350}{1827}",
"style": "rule"
} | {
"index": "3848",
"split": "train"
} |
math | [
{
"content": "The integer $n$ is the largest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\\frac{n}{15}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "592",
"style": "rule"
} | {
"index": "3849",
"split": "train"
} |
math | [
{
"content": "Given that there are three mathematics teachers: Mrs. Germain with 13 students, Mr. Newton with 10 students, and Mrs. Young with 12 students, and 2 students are taking classes from both Mrs. Germain and Mr. Newton and 1 additional student is taking classes from both Mrs. Germain and Mrs. Young. De... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "3850",
"split": "train"
} |
math | [
{
"content": "Find the largest natural number consisting of distinct digits such that the product of its digits equals 2016. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "876321",
"style": "rule"
} | {
"index": "3851",
"split": "train"
} |
math | [
{
"content": "The number of elderly employees in the sampled group is what fraction of the number of young employees in the sampled group if the sampled group consists of the employees of a certain unit, which has a total of 430 employees, among which there are 160 young employees, and the number of middle-aged... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "3852",
"split": "train"
} |
math | [
{
"content": "The function $g(x)$ satisfies\n\\[g(x) - 2 g \\left( \\frac{1}{x} \\right) = 3^x\\] for all \\( x \\neq 0 \\). Find $g(2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{29}{9}",
"style": "rule"
} | {
"index": "3853",
"split": "train"
} |
math | [
{
"content": "Consider a \"Modulo $m$ graph paper,\" with $m = 17$, forming a grid of $17^2$ points. Each point $(x, y)$ represents integer pairs where $0 \\leq x, y < 17$. We graph the congruence $$5x \\equiv 3y + 2 \\pmod{17}.$$ The graph has single $x$-intercept $(x_0, 0)$ and single $y$-intercept $(0, y_0)$... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "3854",
"split": "train"
} |
math | [
{
"content": "Compute $10^{-1}\\pmod{1001}$. Express your answer as a residue from $0$ to $1000$, inclusive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "901",
"style": "rule"
} | {
"index": "3855",
"split": "train"
} |
math | [
{
"content": "Let $\\{a_n\\}$ be an arithmetic sequence. If we select any 4 different numbers from $\\{a_1, a_2, a_3, \\ldots, a_{10}\\}$ such that these 4 numbers still form an arithmetic sequence, then the maximum number of such arithmetic sequences is $\\boxed{24}$. Please output the final answer within \\bo... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "3856",
"split": "train"
} |
math | [
{
"content": "The rightmost nonzero digit in the decimal expansion of 101 ! is the same as the rightmost nonzero digit of $n$ !, where $n$ is an integer greater than 101. Find the smallest possible value of $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "103",
"style": "rule"
} | {
"index": "3857",
"split": "train"
} |
math | [
{
"content": "The Bank of Oslo issues two types of coin: aluminum (denoted A) and bronze (denoted B). Marianne has $n$ aluminum coins and $n$ bronze coins arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k \\leq ... | {
"ground_truth": "{n \\leq k \\leq \\lceil \\tfrac32n \\rceil}",
"style": "rule"
} | {
"index": "3858",
"split": "train"
} |
math | [
{
"content": "Inside triangle $ABC$, there are 1997 points. Using the vertices $A, B, C$ and these 1997 points, into how many smaller triangles can the original triangle be divided? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3995",
"style": "rule"
} | {
"index": "3859",
"split": "train"
} |
math | [
{
"content": "Let a three-digit number \\( n = \\overline{abc} \\), where \\( a \\), \\( b \\), and \\( c \\) can form an isosceles (including equilateral) triangle as the lengths of its sides. How many such three-digit numbers \\( n \\) are there? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "165",
"style": "rule"
} | {
"index": "3860",
"split": "train"
} |
math | [
{
"content": "Two farmers agree that pigs are worth $\\$300$ and that goats are worth $\\$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $\\$390$ debt could be paid with two pigs, with one goat re... | {
"ground_truth": "\\$30",
"style": "rule"
} | {
"index": "3861",
"split": "train"
} |
math | [
{
"content": "A \"fifty percent mirror\" is a mirror that reflects half the light shined on it back and passes the other half of the light onward. Two \"fifty percent mirrors\" are placed side by side in parallel, and a light is shined from the left of the two mirrors. How much of the light is reflected back to... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "3862",
"split": "train"
} |
math | [
{
"content": "Two circles have the same center $C.$ (Circles which have the same center are called concentric.) The larger circle has radius $10$ and the smaller circle has radius $6.$ Determine the area of the ring between these two circles. [asy]\nimport graph;\nfilldraw(circle((0,0),10), lightgray, black+lin... | {
"ground_truth": "64\\pi",
"style": "rule"
} | {
"index": "3863",
"split": "train"
} |
math | [
{
"content": "A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white? Please output the final answer withi... | {
"ground_truth": "\\frac{5}{9}",
"style": "rule"
} | {
"index": "3864",
"split": "train"
} |
math | [
{
"content": "Let \\(x\\) and \\(y\\) be positive real numbers. Find the minimum value of\n\\[\n\\left( x + \\frac{1}{y} \\right) \\left( x + \\frac{1}{y} - 1000 \\right) + \\left( y + \\frac{1}{x} \\right) \\left( y + \\frac{1}{x} - 1000 \\right).\n\\] Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "-500000",
"style": "rule"
} | {
"index": "3865",
"split": "train"
} |
math | [
{
"content": "Express the value of the following expression as a fraction. $$1+\\cfrac{2}{3+\\cfrac{4}{5}}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{29}{19}",
"style": "rule"
} | {
"index": "3866",
"split": "train"
} |
math | [
{
"content": "The line $ax+(a+1)y=a+2$ passes through the point $(4,-8)$. Find $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "3867",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{R}$ be the matrix for reflecting over the vector $\\begin{pmatrix} 3 \\\\ 1 \\end{pmatrix}.$ Find $\\mathbf{R}^2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}",
"style": "rule"
} | {
"index": "3868",
"split": "train"
} |
math | [
{
"content": "Let $f(x)$ be a third-degree polynomial with real coefficients satisfying \\[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\\]Find $|f(0)|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "3869",
"split": "train"
} |
math | [
{
"content": "Find the sum of $543_7$, $65_7$, and $6_7$ in base $7$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "650_7",
"style": "rule"
} | {
"index": "3870",
"split": "train"
} |
math | [
{
"content": "A $n$-gon $S-A_{1} A_{2} \\cdots A_{n}$ has its vertices colored such that each vertex is colored with one color, and the endpoints of each edge are colored differently. Given $n+1$ colors available, find the number of different ways to color the vertices. (For $n=4$, this was a problem in the 199... | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "3871",
"split": "train"
} |
math | [
{
"content": "Calculate $x$ such that the sum \\[1 \\cdot 1979 + 2 \\cdot 1978 + 3 \\cdot 1977 + \\dots + 1978 \\cdot 2 + 1979 \\cdot 1 = 1979 \\cdot 990 \\cdot x.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "661",
"style": "rule"
} | {
"index": "3872",
"split": "train"
} |
math | [
{
"content": "A square is inscribed in the ellipse\n\\[\\frac{x^2}{3} + \\frac{y^2}{6} = 1,\\]so that its sides are parallel to the coordinate axes. Find the area of the square.\n\n[asy]\nunitsize(1 cm);\n\ndraw(xscale(sqrt(3))*yscale(sqrt(6))*Circle((0,0),1));\ndraw((sqrt(2),sqrt(2))--(-sqrt(2),sqrt(2))--(-sq... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "3873",
"split": "train"
} |
math | [
{
"content": "For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $7-3i$ and the sum of the other two roots is $5-2i,$ where $i^2 = -1.$ Find $b.$ Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "43",
"style": "rule"
} | {
"index": "3874",
"split": "train"
} |
math | [
{
"content": "Let \\( a \\) and \\( b \\) be nonnegative real numbers such that\n\\[ \\cos (ax + b) = \\cos 31x \\]\nfor all integers \\( x \\). Find the smallest possible value of \\( a \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "3875",
"split": "train"
} |
math | [
{
"content": "Find the least possible sum of two bases, $c$ and $d$, such that the numeral $29$ in base $c$ represents the same number as $92$ in base $d$, where $c$ and $d$ are positive integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "3876",
"split": "train"
} |
math | [
{
"content": "Given a triangle $ABC$ with $\\angle B = \\frac{\\pi}{3}$, \n(I) if $AB=8\\sqrt{3}$ and $AC=12$, find the area of $\\triangle ABC$; \n(II) if $AB=4$ and $\\vec{BM} = \\vec{MN} = \\vec{NC}$ with $AN=2\\sqrt{3}BM$, find the length of $AM$. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "3877",
"split": "train"
} |
math | [
{
"content": "Find the value of $c$ such that $6x^2 + cx + 16$ equals the square of a binomial. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8\\sqrt{6}",
"style": "rule"
} | {
"index": "3878",
"split": "train"
} |
math | [
{
"content": "Determine the number of six-letter words where the first and last letters are the same, and the second and fifth letters are also the same. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "456976",
"style": "rule"
} | {
"index": "3879",
"split": "train"
} |
math | [
{
"content": "The area of parallelogram $ABCD$ is $51\\sqrt{55}$ and $\\angle{DAC}$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "3880",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of the expression\n\n$$\n\\left(x_{1}-x_{2}\\right)^{2}+\\left(x_{2}-x_{3}\\right)^{2}+\\ldots+\\left(x_{2010}-x_{2011}\\right)^{2}+\\left(x_{2011}-x_{1}\\right)^{2}\n$$\n\nwhere \\(x_{1}, \\ldots, x_{2011} \\in [0, 1]\\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "2010",
"style": "rule"
} | {
"index": "3881",
"split": "train"
} |
math | [
{
"content": "In a polar coordinate system, the midpoint of the line segment whose endpoints are $\\left( 8, \\frac{5 \\pi}{12} \\right)$ and $\\left( 8, -\\frac{3 \\pi}{12} \\right)$ is the point $(r, \\theta).$ Enter $(r, \\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$ Please output the final answer ... | {
"ground_truth": "\\left( 4, \\frac{\\pi}{12} \\right)",
"style": "rule"
} | {
"index": "3882",
"split": "train"
} |
math | [
{
"content": "Given that $\\overset{→}{a\\_n}=\\left(\\cos \\frac{nπ}{6},\\sin \\frac{nπ}{6}\\right)$, $n∈ℕ^∗$, $\\overset{→}{b}=\\left( \\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right)$, calculate the value of $y={\\left| \\overset{→}{{a\\_1}}+ \\overset{→}{b}\\right|}^{2}+{\\left| \\overset{→}{{a\\_2}}+ \\overset{→... | {
"ground_truth": "4029",
"style": "rule"
} | {
"index": "3883",
"split": "train"
} |
math | [
{
"content": "The circular region of the sign (below, left) has an area of 154 square inches. Vanessa would like to place a tiny ribbon (shaded) around the circle's edge. To be sure she has enough ribbon, she decides to buy 2 inches more of the ribbon than the original circle's circumference. How many inches of... | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "3884",
"split": "train"
} |
math | [
{
"content": "Let $P$ be a polynomial with integer coefficients such that $P(0)+P(90)=2018$. Find the least possible value for $|P(20)+P(70)|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\[ 782 \\]",
"style": "rule"
} | {
"index": "3885",
"split": "train"
} |
math | [
{
"content": "If $x = 3$, what is the value of $2x + 3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "3886",
"split": "train"
} |
math | [
{
"content": "Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the highest total score after the three events wins the championship. It is known that the probabilities of school A winning in th... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "3887",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $(xOy)$, a curve $C_{1}$ is defined by the parametric equations $x=\\cos{\\theta}$ and $y=\\sin{\\theta}$, and a line $l$ is defined by the polar equation $\\rho(2\\cos{\\theta} - \\sin{\\theta}) = 6$.\n\n1. Find the Cartesian equations for the curve $C_{1}$ and ... | {
"ground_truth": "\\frac{6\\sqrt{5}}{5} - 1",
"style": "rule"
} | {
"index": "3888",
"split": "train"
} |
math | [
{
"content": "A certain car beauty company launched a promotional activity to attract customers: for first-time customers, the charge is $200 per visit, and they can register as members. Members will receive corresponding discounts for each subsequent visit, as shown in the table below:\n\n| Number of Visits | ... | {
"ground_truth": "\\frac{8}{15}",
"style": "rule"
} | {
"index": "3889",
"split": "train"
} |
math | [
{
"content": "Given that the two roots of the equation $x^{2}+3ax+3a+1=0$ where $a > 1$ are $\\tan \\alpha$ and $\\tan \\beta$, and $\\alpha, \\beta \\in \\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$, find the value of $\\alpha + \\beta$. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "-\\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": "3890",
"split": "train"
} |
math | [
{
"content": "At around 8 o'clock in the morning, two cars left the fertilizer plant one after another, heading toward Happy Village. Both cars travel at a speed of 60 kilometers per hour. At 8:32, the distance the first car had traveled from the fertilizer plant was three times the distance traveled by the sec... | {
"ground_truth": "8:11",
"style": "rule"
} | {
"index": "3891",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\\{1, 2, ..., 30\\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "295",
"style": "rule"
} | {
"index": "3892",
"split": "train"
} |
math | [
{
"content": "The value of $y$ varies inversely as $\\sqrt x$ and when $x=2$, $y=4$. What is $x$ when $y=1$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "3893",
"split": "train"
} |
math | [
{
"content": "There are two islands, A and B, that are 20 nautical miles apart. When viewing Island C from Island A, the angle between Island B and Island C is 60°. When viewing Island C from Island B, the angle between Island A and Island C is 75°. Find the distance between Island B and Island C. Please output... | {
"ground_truth": "10\\sqrt{6}",
"style": "rule"
} | {
"index": "3894",
"split": "train"
} |
math | [
{
"content": "Given that $M$ and $N$ are the common points of circle $A$: $x^{2}+y^{2}-2x=0$ and circle $B$: $x^{2}+y^{2}+2x-4y=0$, find the area of $\\triangle BMN$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "3895",
"split": "train"
} |
math | [
{
"content": "Given that \\(AD\\), \\(BE\\), and \\(CF\\) are the altitudes of the acute triangle \\(\\triangle ABC\\). If \\(AB = 26\\) and \\(\\frac{EF}{BC} = \\frac{5}{13}\\), what is the length of \\(BE\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "3896",
"split": "train"
} |
math | [
{
"content": "In civil engineering, the stability of cylindrical columns under a load is often examined using the formula \\( L = \\frac{50T^3}{SH^2} \\), where \\( L \\) is the load in newtons, \\( T \\) is the thickness of the column in centimeters, \\( H \\) is the height of the column in meters, and \\( S \... | {
"ground_truth": "31.25",
"style": "rule"
} | {
"index": "3897",
"split": "train"
} |
math | [
{
"content": "In a round-robin chess tournament with $x$ players, two players dropped out after playing three matches each. The tournament ended with a total of 84 matches played. How many players were there initially? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "3898",
"split": "train"
} |
math | [
{
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is $S_n$, and it is given that $S_4=20$, $S_{n-4}=60$, and $S_n=120$. Find the value of $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "3899",
"split": "train"
} |
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