data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "What is the greatest possible three-digit number that is divisible by 3 and divisible by 6? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "996",
"style": "rule"
} | {
"index": "6500",
"split": "train"
} |
math | [
{
"content": "At Central Middle School, the $150$ students who participate in the Math Club plan to gather for a game night. Due to an overlapping school event, attendance is expected to drop by $40\\%$. Walter and Gretel are preparing cookies using a new recipe that makes $18$ cookies per batch. The other deta... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "6501",
"split": "train"
} |
math | [
{
"content": "Find all the roots of the polynomial $x^{5}-5 x^{4}+11 x^{3}-13 x^{2}+9 x-3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1, \\frac{3+\\sqrt{3} i}{2}, \\frac{1-\\sqrt{3} i}{2}, \\frac{3-\\sqrt{3} i}{2}, \\frac{1+\\sqrt{3} i}{2}",
"style": "rule"
} | {
"index": "6502",
"split": "train"
} |
math | [
{
"content": "Let $ m\\equal{}\\left(abab\\right)$ and $ n\\equal{}\\left(cdcd\\right)$ be four-digit numbers in decimal system. If $ m\\plus{}n$ is a perfect square, what is the largest value of $ a\\cdot b\\cdot c\\cdot d$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "6503",
"split": "train"
} |
math | [
{
"content": "The average age of seven children is 8 years old. Each child is a different age, and there is a difference of three years in the ages of any two consecutive children. In years, how old is the oldest child? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "6504",
"split": "train"
} |
math | [
{
"content": "A set \\( A \\) consists of 40 elements chosen from \\(\\{1, 2, \\ldots, 50\\}\\), and \\( S \\) is the sum of all elements in the set \\( A \\). How many distinct values can \\( S \\) take? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "401",
"style": "rule"
} | {
"index": "6505",
"split": "train"
} |
math | [
{
"content": "Evaluate the infinite sum $\\sum_{n=0}^{\\infty}\\binom{2 n}{n} \\frac{1}{5^{n}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "6506",
"split": "train"
} |
math | [
{
"content": "(a) A natural number \\( n \\) is less than 120. What is the maximum remainder that the number 209 can leave when divided by \\( n \\)?\n\n(b) A natural number \\( n \\) is less than 90. What is the maximum remainder that the number 209 can leave when divided by \\( n \\)? Please output the final ... | {
"ground_truth": "69",
"style": "rule"
} | {
"index": "6507",
"split": "train"
} |
math | [
{
"content": "A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $\\text{X}$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "Y",
"style": "rule"
} | {
"index": "6508",
"split": "train"
} |
math | [
{
"content": "If $\\tan x+\\tan y=4$ and $\\cot x+\\cot y=5$, compute $\\tan (x+y)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "6509",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=e^{ax}-x-1$, where $a\\neq 0$. If $f(x)\\geqslant 0$ holds true for all $x\\in R$, then the set of possible values for $a$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\{1\\}",
"style": "rule"
} | {
"index": "6510",
"split": "train"
} |
math | [
{
"content": "Given the sample data set $3$, $3$, $4$, $4$, $5$, $6$, $6$, $7$, $7$, calculate the standard deviation of the data set. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{5}}{3}",
"style": "rule"
} | {
"index": "6511",
"split": "train"
} |
math | [
{
"content": "The expression\n\\[\\frac{P+Q}{P-Q}-\\frac{P-Q}{P+Q}\\]\nwhere $P=x+y$ and $Q=x-y$, is equivalent to: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{x^2-y^2}{xy}",
"style": "rule"
} | {
"index": "6512",
"split": "train"
} |
math | [
{
"content": "A solid has a triangular base with sides of lengths $s$, $s$, $s \\sqrt{2}$. Two opposite vertices of the triangle extend vertically upward by a height $h$ where $h = 3s$. Given $s = 2\\sqrt{2}$, what is the volume of this solid? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "24\\sqrt{2}",
"style": "rule"
} | {
"index": "6513",
"split": "train"
} |
math | [
{
"content": "A $2\\times 3$ rectangle and a $3\\times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square? Please output the final answer within ... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "6514",
"split": "train"
} |
math | [
{
"content": "Calculate: $\\sqrt[3]{27}+|-\\sqrt{2}|+2\\sqrt{2}-(-\\sqrt{4})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5 + 3\\sqrt{2}",
"style": "rule"
} | {
"index": "6515",
"split": "train"
} |
math | [
{
"content": "Henry walks $\\tfrac{3}{4}$ of the way from his home to his gym, which is $2$ kilometers away from Henry's home, and then walks $\\tfrac{3}{4}$ of the way from where he is back toward home. Determine the difference in distance between the points toward which Henry oscillates from home and the gym.... | {
"ground_truth": "\\frac{6}{5}",
"style": "rule"
} | {
"index": "6516",
"split": "train"
} |
math | [
{
"content": "What is the value of\n$$\n(x+1)(x+2006)\\left[\\frac{1}{(x+1)(x+2)}+\\frac{1}{(x+2)(x+3)}+\\ldots+\\frac{1}{(x+2005)(x+2006)}\\right] ?\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2005",
"style": "rule"
} | {
"index": "6517",
"split": "train"
} |
math | [
{
"content": "If $\\triangle A_1A_2A_3$ is equilateral and $A_{n+3}$ is the midpoint of line segment $A_nA_{n+1}$ for all positive integers $n$, then the measure of $\\measuredangle A_{44}A_{45}A_{43}$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60^{\\circ}",
"style": "rule"
} | {
"index": "6518",
"split": "train"
} |
math | [
{
"content": "The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?\n$\\begin{array}{c@{}c@{}c@{}c@{}c}\n& & 6 & 4 & 1 \\\\\n& & 8 & 5 & 2 \\\\\n& + & 9 & 7 & 3 \\\\\n\\hline\n& 2 & 4 & 5 & 6\n\\end{array}$ Please output the final answer within \\boxed{}.",... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "6519",
"split": "train"
} |
math | [
{
"content": "Among the numbers $210_{(6)}$, $1000_{(4)}$, and $111111_{(2)}$, the smallest number is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "111111_{(2)}",
"style": "rule"
} | {
"index": "6520",
"split": "train"
} |
math | [
{
"content": "**Q8.** Given a triangle $ABC$ and $2$ point $K \\in AB, \\; N \\in BC$ such that $BK=2AK, \\; CN=2BN$ and $Q$ is the common point of $AN$ and $CK$ . Compute $\\dfrac{ S_{ \\triangle ABC}}{S_{\\triangle BCQ}}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "7/4",
"style": "rule"
} | {
"index": "6521",
"split": "train"
} |
math | [
{
"content": "Find\n\\[\\sum_{n = 1}^\\infty \\frac{2^n}{1 + 2^n + 2^{n + 1} + 2^{2n + 1}}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "6522",
"split": "train"
} |
math | [
{
"content": "If \\( y+4=(x-2)^{2} \\) and \\( x+4=(y-2)^{2} \\), and \\( x \\neq y \\), then the value of \\( x^{2}+y^{2} \\) is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "6523",
"split": "train"
} |
math | [
{
"content": "Let the equation $x^3 - 6x^2 + 11x - 6 = 0$ have three real roots $a$, $b$, and $c$. Find $\\frac{1}{a^3} + \\frac{1}{b^3} + \\frac{1}{c^3}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{251}{216}",
"style": "rule"
} | {
"index": "6524",
"split": "train"
} |
math | [
{
"content": "What is the modulo $5$ remainder of the sum $1+2+3+4+5+ \\ldots + 120+121+122+123?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "6525",
"split": "train"
} |
math | [
{
"content": "Express $0.\\overline{3}+0.\\overline{2}$ as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac 59",
"style": "rule"
} | {
"index": "6526",
"split": "train"
} |
math | [
{
"content": "A square is inscribed in a circle. The number of inches in the perimeter of the square equals the number of square inches in the area of the circumscribed circle. What is the radius, in inches, of the circle? Express your answer in terms of pi. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{4\\sqrt{2}}{\\pi}",
"style": "rule"
} | {
"index": "6527",
"split": "train"
} |
math | [
{
"content": "What is the total number of digits used when the first 2500 positive even integers are written? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9449",
"style": "rule"
} | {
"index": "6528",
"split": "train"
} |
math | [
{
"content": "A projection takes $\\begin{pmatrix} 4 \\\\ 4 \\end{pmatrix}$ to $\\begin{pmatrix} \\frac{60}{13} \\\\ \\frac{12}{13} \\end{pmatrix}.$ Which vector does the projection take $\\begin{pmatrix} -2 \\\\ 2 \\end{pmatrix}$ to? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} -20/13 \\\\ -4/13 \\end{pmatrix}",
"style": "rule"
} | {
"index": "6529",
"split": "train"
} |
math | [
{
"content": "In rectangle $ABCD$, $\\overline{AB}=20$ and $\\overline{BC}=10$. Let $E$ be a point on $\\overline{CD}$ such that $\\angle CBE=15^\\circ$. What is $\\overline{AE}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "6530",
"split": "train"
} |
math | [
{
"content": "Two players, A and B, play a game called \"draw the joker card\". In the beginning, Player A has $n$ different cards. Player B has $n+1$ cards, $n$ of which are the same with the $n$ cards in Player A's hand, and the rest one is a Joker (different from all other $n$ cards). The rules are i) Player... | {
"ground_truth": "n=32",
"style": "rule"
} | {
"index": "6531",
"split": "train"
} |
math | [
{
"content": "If $1+2x+3x^2 + \\dotsb=9$, find $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "6532",
"split": "train"
} |
math | [
{
"content": "Find (in terms of $n \\geq 1$) the number of terms with odd coefficients after expanding the product: $\\prod_{1 \\leq i<j \\leq n}\\left(x_{i}+x_{j}\\right)$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "n!",
"style": "rule"
} | {
"index": "6533",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer that is divisible by the next three distinct primes larger than 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1001",
"style": "rule"
} | {
"index": "6534",
"split": "train"
} |
math | [
{
"content": "Let the function \\( y = f(x) \\) satisfy: for all \\( x \\in \\mathbb{R} \\), \\( y = f(x) \\geqslant 0 \\), and \\( f(x+1) = \\sqrt{9 - f(x)^2} \\). When \\( x \\in [0,1) \\),\n$$\nf(x) = \\begin{cases} \n2^x, & 0 \\leqslant x < \\frac{1}{2}, \\\\\n\\log_{10} (x + 31), & \\frac{1}{2} \\leqslant ... | {
"ground_truth": "\\frac{3\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "6535",
"split": "train"
} |
math | [
{
"content": "The value of $x + x(x^x)$ when $x = 2$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "6536",
"split": "train"
} |
math | [
{
"content": "Find the sum of the coefficients of the polynomial $P(x)=x^{4}-29 x^{3}+a x^{2}+b x+c$, given that $P(5)=11, P(11)=17$, and $P(17)=23$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3193",
"style": "rule"
} | {
"index": "6537",
"split": "train"
} |
math | [
{
"content": "Farmer Bill's 1000 animals - ducks, cows, and rabbits - are standing in a circle. In order to feel safe, every duck must either be standing next to at least one cow or between two rabbits. If there are 600 ducks, what is the least number of cows there can be for this to be possible? Please output ... | {
"ground_truth": "201",
"style": "rule"
} | {
"index": "6538",
"split": "train"
} |
math | [
{
"content": "Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$a+4$",
"style": "rule"
} | {
"index": "6539",
"split": "train"
} |
math | [
{
"content": "The graphs of $2y + x + 3 = 0$ and $3y + ax + 2 = 0$ are perpendicular. Solve for $a.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "6540",
"split": "train"
} |
math | [
{
"content": "An infantry column stretched over 1 km. Sergeant Kim, riding a gyroscooter from the end of the column, reached its front and then returned to the end. During this time, the infantrymen covered 2 km 400 m. What distance did the sergeant travel during this time? Please output the final answer within... | {
"ground_truth": "3.6",
"style": "rule"
} | {
"index": "6541",
"split": "train"
} |
math | [
{
"content": "Determine the volume in cubic inches of a right, rectangular prism where the areas of the side, front, and bottom faces are 20 square inches, 12 square inches, and 8 square inches, respectively. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8\\sqrt{30}",
"style": "rule"
} | {
"index": "6542",
"split": "train"
} |
math | [
{
"content": "Given $S$ is the set of the 1000 smallest positive multiples of $5$, and $T$ is the set of the 1000 smallest positive multiples of $9$, determine the number of elements common to both sets $S$ and $T$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "111",
"style": "rule"
} | {
"index": "6543",
"split": "train"
} |
math | [
{
"content": "Evaluate the polynomial \\[ x^3 - 2 x^2 - 8 x + 4, \\]where $x$ is the positive number such that $x^2 - 2x - 8 = 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "6544",
"split": "train"
} |
math | [
{
"content": "How many vertical asymptotes does the equation $y=\\frac{x-1}{x^2+6x-7}$ have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "6545",
"split": "train"
} |
math | [
{
"content": "Given Ricardo has $3000$ coins comprised of pennies ($1$-cent coins), nickels ($5$-cent coins), and dimes ($10$-cent coins), with at least one of each type of coin, calculate the difference in cents between the highest possible and lowest total value that Ricardo can have. Please output the final ... | {
"ground_truth": "26973",
"style": "rule"
} | {
"index": "6546",
"split": "train"
} |
math | [
{
"content": "Given that $\\sqrt {3} \\overrightarrow{a}+ \\overrightarrow{b}+2 \\overrightarrow{c}= \\overrightarrow{0}$, and $| \\overrightarrow{a}|=| \\overrightarrow{b}|=| \\overrightarrow{c}|=1$, find the value of $\\overrightarrow{a}\\cdot ( \\overrightarrow{b}+ \\overrightarrow{c})$. Please output the fi... | {
"ground_truth": "-\\dfrac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "6547",
"split": "train"
} |
math | [
{
"content": "In the arithmetic sequence $\\{a_n\\}$, it is given that $a_1 > 0$, $a_{2012} + a_{2013} > 0$, and $a_{2012} \\cdot a_{2013} < 0$, calculate the largest natural number n that satisfies S_n > 0. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4024",
"style": "rule"
} | {
"index": "6548",
"split": "train"
} |
math | [
{
"content": "In a math competition with problems $A$, $B$, and $C$, there are 39 participants, each of whom answered at least one question correctly. Among those who answered problem $A$ correctly, the number of participants who answered only problem $A$ is 5 more than those who also answered other problems. A... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "6549",
"split": "train"
} |
math | [
{
"content": "Several positive integers are given, not necessarily all different. Their sum is 2003. Suppose that $n_{1}$ of the given numbers are equal to $1, n_{2}$ of them are equal to $2, \\ldots, n_{2003}$ of them are equal to 2003. Find the largest possible value of $$n_{2}+2 n_{3}+3 n_{4}+\\cdots+2002 n_... | {
"ground_truth": "2002",
"style": "rule"
} | {
"index": "6550",
"split": "train"
} |
math | [
{
"content": "A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not include the area beneath the c... | {
"ground_truth": "166",
"style": "rule"
} | {
"index": "6551",
"split": "train"
} |
math | [
{
"content": "A number is chosen at random from the set of consecutive natural numbers $\\{1, 2, 3, \\ldots, 24\\}$. What is the probability that the number chosen is a factor of $4!$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "6552",
"split": "train"
} |
math | [
{
"content": "Find the smallest two-digit number \\( N \\) such that the sum of digits of \\( 10^N - N \\) is divisible by 170. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "6553",
"split": "train"
} |
math | [
{
"content": "Let $S$ denote the value of the sum\\[\\sum_{n=0}^{668} (-1)^{n} {2004 \\choose 3n}\\]Determine the remainder obtained when $S$ is divided by $1000$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "6554",
"split": "train"
} |
math | [
{
"content": "The lines $y=x, y=2 x$, and $y=3 x$ are the three medians of a triangle with perimeter 1. Find the length of the longest side of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{\\frac{\\sqrt{58}}{2+\\sqrt{34}+\\sqrt{58}}}",
"style": "rule"
} | {
"index": "6555",
"split": "train"
} |
math | [
{
"content": "There are 11 ones, 22 twos, 33 threes, and 44 fours on a blackboard. The following operation is performed: each time, erase 3 different numbers and add 2 more of the fourth number that was not erased. For example, if 1 one, 1 two, and 1 three are erased in one operation, write 2 more fours. After ... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "6556",
"split": "train"
} |
math | [
{
"content": "Given a geometric sequence $\\{a_n\\}$, where $a_3$ and $a_7$ are the two roots of the quadratic equation $x^2+7x+9=0$, calculate the value of $a_5$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "6557",
"split": "train"
} |
math | [
{
"content": "What is the value of $((-3)^2)^4(-3)^8 \\cdot 2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "86093442",
"style": "rule"
} | {
"index": "6558",
"split": "train"
} |
math | [
{
"content": "The graph of a certain quadratic $y = ax^2 + bx + c$ is a parabola with vertex $(-4,0)$ which passes through the point $(1,-75)$. What is the value of $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "6559",
"split": "train"
} |
math | [
{
"content": "Regular polygons $I C A O, V E N T I$, and $A L B E D O$ lie on a plane. Given that $I N=1$, compute the number of possible values of $O N$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "6560",
"split": "train"
} |
math | [
{
"content": "There are $2022$ users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.) \n\nStarting now, Mathbook will only allow a new friendship to be formed between two users if they have [i]at least two[/i] friends in common... | {
"ground_truth": "3031",
"style": "rule"
} | {
"index": "6561",
"split": "train"
} |
math | [
{
"content": "Let $L,E,T,M,$ and $O$ be digits that satisfy $LEET+LMT=TOOL.$ Given that $O$ has the value of $0,$ digits may be repeated, and $L\\neq0,$ what is the value of the $4$ -digit integer $ELMO?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1880",
"style": "rule"
} | {
"index": "6562",
"split": "train"
} |
math | [
{
"content": "The average age of $5$ people in a room is $30$ years. An $18$-year-old person leaves the room. What is the average age of the four remaining people? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "6563",
"split": "train"
} |
math | [
{
"content": "Find the 6-digit repetend in the decimal representation of $\\frac 3{13}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "230769",
"style": "rule"
} | {
"index": "6564",
"split": "train"
} |
math | [
{
"content": "Given that $x=1$ is a root of the quadratic equation $\\left(m-2\\right)x^{2}+4x-m^{2}=0$, calculate the value(s) of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "6565",
"split": "train"
} |
math | [
{
"content": "Arrange 3 boys and 4 girls in a row. Calculate the number of different arrangements under the following conditions:\n(1) Person A and Person B must stand at the two ends; \n(2) All boys must stand together; \n(3) No two boys stand next to each other; \n(4) Exactly one person stands between Pers... | {
"ground_truth": "1200",
"style": "rule"
} | {
"index": "6566",
"split": "train"
} |
math | [
{
"content": "The positive five-digit integers that use each of the five digits $1,$ $2,$ $3,$ $4,$ and $5$ exactly once are ordered from least to greatest. What is the $50^{\\text{th}}$ integer in the list? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31254",
"style": "rule"
} | {
"index": "6567",
"split": "train"
} |
math | [
{
"content": "Given that Star lists the whole numbers $1$ through $30$ once and Emilio copies those numbers replacing each occurrence of the digit $3$ by the digit $2$, find the difference between Star's and Emilio's total sums. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "6568",
"split": "train"
} |
math | [
{
"content": "The eight digits in Jane's phone number and the four digits in her office number have the same sum. The digits in her office number are distinct, and her phone number is 539-8271. What is the largest possible value of Jane's office number? Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "9876",
"style": "rule"
} | {
"index": "6569",
"split": "train"
} |
math | [
{
"content": "Given that right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\\overline{AC}$ of isosceles right triangle $ABC$ with leg length $2$, and $\\angle CAD = 30^{\\circ}$, find $\\sin(2\\angle BAD)$. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "6570",
"split": "train"
} |
math | [
{
"content": "Let $a_{0},a_{1},a_{2},\\ldots $ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i,j$ and $k$ are not necessarily distinct. Determine $a_{1998}$. Please output the final answer within \\boxed{... | {
"ground_truth": "{11111001110_8}",
"style": "rule"
} | {
"index": "6571",
"split": "train"
} |
math | [
{
"content": "Two fair, eight-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 12? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{49}{64}",
"style": "rule"
} | {
"index": "6572",
"split": "train"
} |
math | [
{
"content": "Let $\\Delta ABC$ be an equilateral triangle. How many squares in the same plane as $\\Delta ABC$ share two vertices with the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "6573",
"split": "train"
} |
math | [
{
"content": "Given real numbers \\( x_1, x_2, \\ldots, x_{2021} \\) satisfy \\( \\sum_{i=1}^{2021} x_i^2 = 1 \\), find the maximum value of \\( \\sum_{i=1}^{2020} x_i^3 x_{i+1}^3 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "6574",
"split": "train"
} |
math | [
{
"content": "Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\\tfrac{1}{2}$, independently of wh... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "6575",
"split": "train"
} |
math | [
{
"content": "Determine the number of subsets $S$ of $\\{1,2, \\ldots, 1000\\}$ that satisfy the following conditions: - $S$ has 19 elements, and - the sum of the elements in any non-empty subset of $S$ is not divisible by 20 . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8 \\cdot\\binom{50}{19}",
"style": "rule"
} | {
"index": "6576",
"split": "train"
} |
math | [
{
"content": "Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly two letters are repeated once each (meaning two pairs of identical letters), and digits can be repeated? Please output the f... | {
"ground_truth": "390000",
"style": "rule"
} | {
"index": "6577",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $AB = AC = 100$, and $BC = 56$. Circle $P$ has radius $16$ and is tangent to $\\overline{AC}$ and $\\overline{BC}$. Circle $Q$ is externally tangent to $P$ and is tangent to $\\overline{AB}$ and $\\overline{BC}$. No point of circle $Q$ lies outside of $\\triangle ABC$. The radiu... | {
"ground_truth": "254",
"style": "rule"
} | {
"index": "6578",
"split": "train"
} |
math | [
{
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is denoted as $S_n$. Given that $a_1-a_5-a_10-a_15+a_19=2$, find the value of $S_{19}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-38",
"style": "rule"
} | {
"index": "6579",
"split": "train"
} |
math | [
{
"content": "Given that the complex number $z_1$ corresponds to the point $(-1,1)$ on the complex plane, and the complex number $z_2$ satisfies $z_1z_2=-2$, find the value of $|z_2+2i|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{10}",
"style": "rule"
} | {
"index": "6580",
"split": "train"
} |
math | [
{
"content": "Given Professor Lewis has twelve different language books lined up on a bookshelf: three Arabic, four German, three Spanish, and two French, calculate the number of ways to arrange the twelve books on the shelf keeping the Arabic books together, the Spanish books together, and the French books tog... | {
"ground_truth": "362,880",
"style": "rule"
} | {
"index": "6581",
"split": "train"
} |
math | [
{
"content": "A builder has two identical bricks. She places them side by side in three different ways, resulting in shapes with surface areas of 72, 96, and 102. What is the surface area of one original brick? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "6582",
"split": "train"
} |
math | [
{
"content": "On a circle, 2009 numbers are placed, each of which is equal to 1 or -1. Not all numbers are the same. Consider all possible groups of ten consecutive numbers. Find the product of the numbers in each group of ten and sum them up. What is the largest possible sum? Please output the final answer wit... | {
"ground_truth": "2005",
"style": "rule"
} | {
"index": "6583",
"split": "train"
} |
math | [
{
"content": "A basement has a 24-foot by 32-foot rectangular floor. The basement is flooded with water to a depth of 18 inches. Three pumps are used to pump the water out of the basement. Each pump will pump 8 gallons of water per minute. If a cubic foot of water contains 7.5 gallons, how many minutes will ... | {
"ground_truth": "360",
"style": "rule"
} | {
"index": "6584",
"split": "train"
} |
math | [
{
"content": "One of the eight faces of a hexagonal prism will be transformed into the base of a new pyramid. Calculate the maximum sum of the number of exterior faces, vertices, and edges of the resultant composite shape (combining the prism and pyramid) when the pyramid is added to each type of face of the pr... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "6585",
"split": "train"
} |
math | [
{
"content": "Rodney is trying to guess a secret number based on the following clues:\n1. It is a two-digit integer.\n2. The tens digit is odd.\n3. The units digit is one of the following: 2, 4, 6, 8.\n4. The number is greater than 50.\n\nWhat is the probability that Rodney will guess the correct number? Expres... | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "6586",
"split": "train"
} |
math | [
{
"content": "Calculate\n\n$$\n\\int_{L}\\left(z^{2}+2z \\bar{z}\\right)d z\n$$\n\nwhere \\( L \\) is the arc of the circle \\( |z|=1 \\) with \\( \\arg z \\in[0, \\pi] \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{14}{3}",
"style": "rule"
} | {
"index": "6587",
"split": "train"
} |
math | [
{
"content": "Given an equilateral triangle of side 10, divide each side into three equal parts, construct an equilateral triangle on the middle part, and then delete the middle part. Repeat this step for each side of the resulting polygon. Find \\( S^2 \\), where \\( S \\) is the area of the region obtained by... | {
"ground_truth": "4800",
"style": "rule"
} | {
"index": "6588",
"split": "train"
} |
math | [
{
"content": "Circles $C_{1}, C_{2}, C_{3}$ have radius 1 and centers $O, P, Q$ respectively. $C_{1}$ and $C_{2}$ intersect at $A, C_{2}$ and $C_{3}$ intersect at $B, C_{3}$ and $C_{1}$ intersect at $C$, in such a way that $\\angle A P B=60^{\\circ}, \\angle B Q C=36^{\\circ}$, and $\\angle C O A=72^{\\circ}$. ... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "6589",
"split": "train"
} |
math | [
{
"content": "Choose $4-4$: Parameter Equation Lecture. In the plane rectangular coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, establish a polar coordinate system. The polar coordinates of point $P$ are $(2\\sqrt{3}, \\dfrac{\\pi}{6})$. The polar coordina... | {
"ground_truth": "\\dfrac{11\\sqrt{5}}{10} - 1",
"style": "rule"
} | {
"index": "6590",
"split": "train"
} |
math | [
{
"content": "From six balls numbered respectively $1$, $2$, $3$, $4$, $5$, $6$, which are all of the same size, three balls are randomly drawn. Find the probability that exactly two of the balls have consecutive numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{3}{5}",
"style": "rule"
} | {
"index": "6591",
"split": "train"
} |
math | [
{
"content": "A line passing through the vertex \\( A \\) of triangle \\( ABC \\) perpendicular to its median \\( BD \\) bisects this median.\n\nFind the ratio of the sides \\( AB \\) and \\( AC \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "6592",
"split": "train"
} |
math | [
{
"content": "There are three flavors of chocolates in a jar: hazelnut, liquor, and milk. There are 12 chocolates that are not hazelnut, 18 chocolates that are not liquor, and 20 chocolates that are not milk. How many chocolates are there in total in the jar? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "6593",
"split": "train"
} |
math | [
{
"content": "Let \\( n = 2^{40}5^{15} \\). How many positive integer divisors of \\( n^2 \\) are less than \\( n \\) but do not divide \\( n \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "599",
"style": "rule"
} | {
"index": "6594",
"split": "train"
} |
math | [
{
"content": "Two students, A and B, are preparing to have a table tennis match during their physical education class. Assuming that the probability of A winning against B in each game is $\\frac{1}{3}$, the match follows a best-of-three format (the first player to win two games wins the match). What is the pro... | {
"ground_truth": "\\frac{7}{27}",
"style": "rule"
} | {
"index": "6595",
"split": "train"
} |
math | [
{
"content": "There exist vectors $\\mathbf{a}$ and $\\mathbf{b}$ such that\n\\[\\mathbf{a} + \\mathbf{b} = \\begin{pmatrix} 6 \\\\ -3 \\\\ -6 \\end{pmatrix},\\]where $\\mathbf{a}$ is parallel to $\\begin{pmatrix} 1 \\\\ 1 \\\\ 1 \\end{pmatrix},$ and $\\mathbf{b}$ is orthogonal to $\\begin{pmatrix} 1 \\\\ 1 \\\... | {
"ground_truth": "\\begin{pmatrix} 7 \\\\ -2 \\\\ -5 \\end{pmatrix}",
"style": "rule"
} | {
"index": "6596",
"split": "train"
} |
math | [
{
"content": "Let $P$ be a point not on line $XY$, and $Q$ a point on line $XY$ such that $PQ \\perp XY$. Meanwhile, $R$ is a point on line $PY$ such that $XR \\perp PY$. Given $XR = 6$, $PQ = 12$, and $XY = 7$, find the length of $PY$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "6597",
"split": "train"
} |
math | [
{
"content": "Given a triangular prism \\( S-ABC \\) with a base that is an isosceles right triangle with \\( AB \\) as the hypotenuse, and \\( SA = SB = SC = AB = 2 \\). If the points \\( S, A, B, C \\) all lie on the surface of a sphere centered at \\( O \\), what is the surface area of this sphere? Please ou... | {
"ground_truth": "\\frac{16 \\pi}{3}",
"style": "rule"
} | {
"index": "6598",
"split": "train"
} |
math | [
{
"content": "A and B plan to meet between 8:00 and 9:00 in the morning, and they agreed that the person who arrives first will wait for the other for 10 minutes before leaving on their own. Calculate the probability that they successfully meet. Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "\\dfrac{11}{36}",
"style": "rule"
} | {
"index": "6599",
"split": "train"
} |
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