data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Find the number of eight-digit positive integers that are multiples of 9 and have all distinct digits. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "181440",
"style": "rule"
} | {
"index": "35400",
"split": "train"
} |
math | [
{
"content": "For all non-zero real numbers $x$ and $y$ such that $x-y=xy$, $\\frac{1}{x}-\\frac{1}{y}$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "35401",
"split": "train"
} |
math | [
{
"content": "$ABCD$ is a rectangular sheet of paper. Points $E$ and $F$ are located on edges $AB$ and $CD$, respectively, such that $BE < CF$. The rectangle is folded over line $EF$ so that point $C$ maps to $C'$ on side $AD$ and point $B$ maps to $B'$ on side $AD$ such that $\\angle{AB'C'} \\cong \\angle{B'EA... | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "35402",
"split": "train"
} |
math | [
{
"content": "Let $\\alpha$ and $\\beta$ be conjugate complex numbers such that $\\frac{\\alpha}{\\beta^3}$ is a real number and $|\\alpha - \\beta| = 6$. Find $|\\alpha|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{2}",
"style": "rule"
} | {
"index": "35403",
"split": "train"
} |
math | [
{
"content": "When Scott completes the square on the quadratic $x^2 + 8x - 1 = 0$, he obtains an equation of the form $(x + a)^2 = b$. What is $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "35404",
"split": "train"
} |
math | [
{
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence, if $\\frac{a_{11}}{a_{10}} < -1$ and its sum of the first $n$ terms, $S_n$, has a maximum value, find the value of $n$ when $S_n$ takes the minimum positive value. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "35405",
"split": "train"
} |
math | [
{
"content": "A rental company owns 100 cars. When the monthly rent for each car is set at 3000 yuan, all cars can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan per month, and for eac... | {
"ground_truth": "4050",
"style": "rule"
} | {
"index": "35406",
"split": "train"
} |
math | [
{
"content": "What is the largest $2$-digit prime factor of the integer $n = {200\\choose 100}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "61",
"style": "rule"
} | {
"index": "35407",
"split": "train"
} |
math | [
{
"content": "A swimming pool can be filled by any of three hoses A, B or C. Hoses A and B together take 4 hours to fill the pool. Hoses A and C together take 5 hours to fill the pool. Hoses B and C together take 6 hours to fill the pool. How many hours does it take hoses A, B and C working together to fill the... | {
"ground_truth": "3.24",
"style": "rule"
} | {
"index": "35408",
"split": "train"
} |
math | [
{
"content": "Ella walks to her university library, averaging 80 steps per minute, with each of her steps covering 80 cm. It takes her 20 minutes to get to the library. Her friend Tia, going to the same library by the same route, averages 120 steps per minute, but her steps are only 70 cm long. Calculate the ti... | {
"ground_truth": "15.24",
"style": "rule"
} | {
"index": "35409",
"split": "train"
} |
math | [
{
"content": "There are four identical balls numbered $1$, $2$, $3$, $4$, and four boxes also numbered $1$, $2$, $3$, $4$.\n\n$(1)$ If each box contains one ball, find the number of ways such that exactly one box has the same number as the ball inside it.\n\n$(2)$ Find the number of ways such that exactly one b... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "35410",
"split": "train"
} |
math | [
{
"content": "Circles \\(P\\), \\(Q\\), and \\(R\\) are externally tangent to each other and internally tangent to circle \\(S\\). Circles \\(Q\\) and \\(R\\) are congruent. Circle \\(P\\) has radius 2 and passes through the center of \\(S\\). What is the radius of circle \\(Q\\)? Please output the final answer... | {
"ground_truth": "\\frac{16}{9}",
"style": "rule"
} | {
"index": "35411",
"split": "train"
} |
math | [
{
"content": "Given that $\\{a_{n}\\}$ is a geometric sequence, and $a_{3}$ and $a_{7}$ are two roots of the equation $x^{2}+4x+1=0$, calculate the value of $a_{5}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "35412",
"split": "train"
} |
math | [
{
"content": "How many times does the digit 9 appear in the list of all integers from 1 to 1000? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "35413",
"split": "train"
} |
math | [
{
"content": "A circle inscribed in a square has two chords as shown in a pair. It has radius 2, and $P$ bisects $T U$. The chords' intersection is where? Answer the question by giving the distance of the point of intersection from the center of the circle. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "2\\sqrt{2} - 2",
"style": "rule"
} | {
"index": "35414",
"split": "train"
} |
math | [
{
"content": "What is the maximum possible product of three different numbers from the set $\\{-9, -7, -2, 0, 4, 6, 8\\}$, where the product contains exactly one negative number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-96",
"style": "rule"
} | {
"index": "35415",
"split": "train"
} |
math | [
{
"content": "For how many integers $x$ is the number $x^4-51x^2+50$ negative? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35416",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\left\\lfloor \\left\\lceil \\left(\\frac{13}{7}\\right)^2\\right\\rceil+\\frac{17}{4}\\right\\rfloor$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "35417",
"split": "train"
} |
math | [
{
"content": "What is $2.43-1.2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.23",
"style": "rule"
} | {
"index": "35418",
"split": "train"
} |
math | [
{
"content": "In writing the integers from 100 through 199 inclusive, how many times is the digit 7 written? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "35419",
"split": "train"
} |
math | [
{
"content": "The closed curve in the figure is made up of 9 congruent circular arcs each of length $\\frac{2\\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve? Please output the final answer within \\box... | {
"ground_truth": "\\pi + 6\\sqrt{3}",
"style": "rule"
} | {
"index": "35420",
"split": "train"
} |
math | [
{
"content": "A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length $4$. A plane passes through the midpoints of $AE$, $BC$, and $CD$. The plane's intersection with the pyramid has an area that can be expressed as $\\sqrt{p}$. Find $p$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "35421",
"split": "train"
} |
math | [
{
"content": "Let $f$ be a monic cubic polynomial satisfying $f(x)+f(-x)=0$ for all real numbers $x$. For all real numbers $y$, define $g(y)$ to be the number of distinct real solutions $x$ to the equation $f(f(x))=y$. Suppose that the set of possible values of $g(y)$ over all real numbers $y$ is exactly $\\{1,... | {
"ground_truth": "970",
"style": "rule"
} | {
"index": "35422",
"split": "train"
} |
math | [
{
"content": "Consider a matrix $A = \\begin{pmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{pmatrix}$ where $a_{11}, a_{12}, a_{21}, a_{22} \\in \\{0, 1\\}$, and the determinant of $A$ is 0. Determine the number of distinct matrices $A$. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "35423",
"split": "train"
} |
math | [
{
"content": "Express $(5-4i)-2(3+6i)$ as a complex number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1-16i",
"style": "rule"
} | {
"index": "35424",
"split": "train"
} |
math | [
{
"content": "A secret facility is a rectangle measuring $200 \\times 300$ meters. Outside the facility, there is one guard at each of the four corners. An intruder approached the perimeter of the facility from the outside, and all the guards ran towards the intruder using the shortest paths along the external ... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "35425",
"split": "train"
} |
math | [
{
"content": "Given the Cartesian coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, a curve $C$ has the polar equation $ρ^2 - 4ρ\\sinθ + 3 = 0$. Points $A$ and $B$ have polar coordinates $(1,π)$ and $(1,0)$, respectively.\n(1) Find the parametric equati... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "35426",
"split": "train"
} |
math | [
{
"content": "The positive integer \\( N \\) is the smallest one whose digits add to 41. What is the sum of the digits of \\( N + 2021 \\)? \n\nA) 10\n\nB) 12\n\nC) 16\n\nD) 2021\n\nE) 4042 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "35427",
"split": "train"
} |
math | [
{
"content": "How many different 4-edge trips are there from $A$ to $B$ in a cube, where the trip can visit one vertex twice (excluding start and end vertices)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "35428",
"split": "train"
} |
math | [
{
"content": "The legs of a right triangle are 3 and 4. Find the area of the triangle with vertices at the points of tangency of the incircle with the sides of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6/5",
"style": "rule"
} | {
"index": "35429",
"split": "train"
} |
math | [
{
"content": "Compute $e^{\\pi}+\\pi^e$ . If your answer is $A$ and the correct answer is $C$ , then your score on this problem will be $\\frac{4}{\\pi}\\arctan\\left(\\frac{1}{\\left|C-A\\right|}\\right)$ (note that the closer you are to the right answer, the higher your score is).\n\n*2017 CCA Math Bona... | {
"ground_truth": "45.5999",
"style": "rule"
} | {
"index": "35430",
"split": "train"
} |
math | [
{
"content": "Find the largest solution to \\[\\lfloor x \\rfloor = 5 + 100 \\{ x \\},\\]where $\\{x\\} = x - \\lfloor x \\rfloor.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "104.99",
"style": "rule"
} | {
"index": "35431",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer $n$ such that $3n \\equiv 1356 \\pmod{22}?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35432",
"split": "train"
} |
math | [
{
"content": "The scent of blooming lily of the valley bushes spreads within a radius of 20 meters around them. How many blooming lily of the valley bushes need to be planted along a straight 400-meter-long alley so that every point along the alley can smell the lily of the valley? Please output the final answe... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "35433",
"split": "train"
} |
math | [
{
"content": "Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\\ne0$ for $0\\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = \\begin{cases}\n\\frac{2}{2 - x}, & 0 \\leq x < 2, \\\\\n0, & x \\geq 2.\n\\end{cases}",
"style": "rule"
} | {
"index": "35434",
"split": "train"
} |
math | [
{
"content": "Given positive numbers $x$ and $y$ satisfying $2x+y=2$, the minimum value of $\\frac{1}{x}-y$ is achieved when $x=$ ______, and the minimum value is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2}-2",
"style": "rule"
} | {
"index": "35435",
"split": "train"
} |
math | [
{
"content": "How many ordered triples \\((x, y, z)\\) satisfy the following conditions:\n\\[ x^2 + y^2 + z^2 = 9, \\]\n\\[ x^4 + y^4 + z^4 = 33, \\]\n\\[ xyz = -4? \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35436",
"split": "train"
} |
math | [
{
"content": "Arrange the sequence $\\{2n+1\\}$ ($n\\in\\mathbb{N}^*$), sequentially in brackets such that the first bracket contains one number, the second bracket two numbers, the third bracket three numbers, the fourth bracket four numbers, the fifth bracket one number, and so on in a cycle: $(3)$, $(5, 7)$,... | {
"ground_truth": "403",
"style": "rule"
} | {
"index": "35437",
"split": "train"
} |
math | [
{
"content": "In how many ways can one arrange the natural numbers from 1 to 9 in a $3 \\times 3$ square table so that the sum of the numbers in each row and each column is odd? (Numbers can repeat) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6 * 4^6 * 5^3 + 9 * 4^4 * 5^5 + 5^9",
"style": "rule"
} | {
"index": "35438",
"split": "train"
} |
math | [
{
"content": "(1) Given the complex number $z=3+bi$ ($i$ is the imaginary unit, $b$ is a positive real number), and $(z-2)^{2}$ is a pure imaginary number, find the complex number $z$;\n(2) Given that the sum of all binomial coefficients in the expansion of $(3x+ \\frac{1}{ \\sqrt{x}})^{n}$ is $16$, find the co... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "35439",
"split": "train"
} |
math | [
{
"content": "Given that five boys, A, B, C, D, and E, are randomly assigned to stay in 3 standard rooms (with at most two people per room), calculate the probability that A and B stay in the same standard room. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "35440",
"split": "train"
} |
math | [
{
"content": "Of the 100 students in the science club, 65 take mathematics, 43 take physics and 10 students take both mathematics and physics. How many science club students take neither mathematics nor physics? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "35441",
"split": "train"
} |
math | [
{
"content": "In a regular hexagon \\(ABCDEF\\), points \\(M\\) and \\(K\\) are taken on the diagonals \\(AC\\) and \\(CE\\) respectively, such that \\(AM : AC = CK : CE = n\\). Points \\(B, M,\\) and \\(K\\) are collinear. Find \\(n\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "35442",
"split": "train"
} |
math | [
{
"content": "Two circles lie outside regular hexagon $ABCDEF$. The first is tangent to $\\overline{AB}$, and the second is tangent to $\\overline{DE}$. Both are tangent to lines $BC$ and $FA$. What is the ratio of the area of the second circle to that of the first circle? Please output the final answer within ... | {
"ground_truth": "81",
"style": "rule"
} | {
"index": "35443",
"split": "train"
} |
math | [
{
"content": "What is the least positive multiple of 45 for which the product of its digits is also a positive multiple of 45? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "945",
"style": "rule"
} | {
"index": "35444",
"split": "train"
} |
math | [
{
"content": "When flipping a fair coin, what is the probability that the first two flips are both heads? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "35445",
"split": "train"
} |
math | [
{
"content": "The first row of a triangle is given as:\n$$\n1, \\frac{1}{2}, \\frac{1}{3}, \\frac{1}{4}, \\ldots, \\frac{1}{1993}\n$$\n\nEach element of the following rows is calculated as the difference between two elements that are above it. The 1993rd row contains only one element. Find this element. Please ... | {
"ground_truth": "\\frac{1}{1993}",
"style": "rule"
} | {
"index": "35446",
"split": "train"
} |
math | [
{
"content": "The sequence $\\{a_n\\}$ satisfies $a_1 = 1$ and $5^{a_{n + 1} - a_n} - 1 = \\frac {1}{n + \\frac {2}{3}}$ for $n \\geq 1$. Find the least integer $k$ greater than $1$ for which $a_k$ is an integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "35447",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\{a_n\\}$ with the sum of its first $n$ terms $S_n = 6n - n^2$, find the sum of the first $20$ terms of the sequence $\\left\\{ \\frac{1}{a_na_{n+1}}\\right\\}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{4}{35}",
"style": "rule"
} | {
"index": "35448",
"split": "train"
} |
math | [
{
"content": "Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$ .\n\n*Proposed by Andy Xu* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "35449",
"split": "train"
} |
math | [
{
"content": "Given $\\triangle ABC$ with the sides opposite to angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, and it satisfies $\\frac {\\sin (2A+B)}{\\sin A}=2+2\\cos (A+B)$. \n(I) Find the value of $\\frac {b}{a}$; \n(II) If $a=1$ and $c= \\sqrt {7}$, find the area of $\\triangle ABC$. Please outpu... | {
"ground_truth": "\\frac { \\sqrt {3}}{2}",
"style": "rule"
} | {
"index": "35450",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "35451",
"split": "train"
} |
math | [
{
"content": "There is a card game called \"Twelve Months\" that is played only during the Chinese New Year. The rules are as follows:\n\nStep 1: Take a brand new deck of playing cards, remove the two jokers and the four Kings, leaving 48 cards. Shuffle the remaining cards.\n\nStep 2: Lay out the shuffled cards... | {
"ground_truth": "1/12",
"style": "rule"
} | {
"index": "35452",
"split": "train"
} |
math | [
{
"content": "In a right triangle $ABC$ (right angle at $C$), the bisector $BK$ is drawn. Point $L$ is on side $BC$ such that $\\angle C K L = \\angle A B C / 2$. Find $KB$ if $AB = 18$ and $BL = 8$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35453",
"split": "train"
} |
math | [
{
"content": "Distribute 4 college students to three factories A, B, and C for internship activities. Factory A can only arrange for 1 college student, the other factories must arrange for at least 1 student each, and student A cannot be assigned to factory C. The number of different distribution schemes is ___... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35454",
"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": "35455",
"split": "train"
} |
math | [
{
"content": "Twenty percent less than 60 is one-third more than what number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "35456",
"split": "train"
} |
math | [
{
"content": "Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "35457",
"split": "train"
} |
math | [
{
"content": "Find all real numbers $x,y,z\\geq 1$ satisfying \\[\\min(\\sqrt{x+xyz},\\sqrt{y+xyz},\\sqrt{z+xyz})=\\sqrt{x-1}+\\sqrt{y-1}+\\sqrt{z-1}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\[\n\\boxed{\\left(\\frac{c^2+c-1}{c^2}, \\frac{c}{c-1}, c\\right)}\n\\]",
"style": "rule"
} | {
"index": "35458",
"split": "train"
} |
math | [
{
"content": "Consider a right rectangular prism \\(B\\) with edge lengths \\(2,\\ 5,\\) and \\(6\\), including its interior. For any real \\(r \\geq 0\\), let \\(T(r)\\) be the set of points in 3D space within a distance \\(r\\) from some point in \\(B\\). The volume of \\(T(r)\\) is expressed as \\(ar^{3} + b... | {
"ground_truth": "\\frac{8112}{240}",
"style": "rule"
} | {
"index": "35459",
"split": "train"
} |
math | [
{
"content": "Calculate\n\n$$\n\\operatorname{tg} \\frac{\\pi}{43} \\cdot \\operatorname{tg} \\frac{2 \\pi}{43}+\\operatorname{tg} \\frac{2 \\pi}{43} \\cdot \\operatorname{tg} \\frac{3 \\pi}{43}+\\ldots+\\operatorname{tg} \\frac{k \\pi}{43} \\cdot \\operatorname{tg} \\frac{(k+1) \\pi}{43}+\\ldots+\\operatorname... | {
"ground_truth": "-2021",
"style": "rule"
} | {
"index": "35460",
"split": "train"
} |
math | [
{
"content": "In the expansion of \\((x + y + z)^8\\), determine the sum of the coefficients of all terms of the form \\(x^2 y^a z^b\\) (\\(a, b \\in \\mathbf{N}\\)). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1792",
"style": "rule"
} | {
"index": "35461",
"split": "train"
} |
math | [
{
"content": "At what value of $a$ do the graphs of $y=ax^2+3x+1$ and $y=-x-1$ intersect at exactly one point? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "35462",
"split": "train"
} |
math | [
{
"content": "The graph of the function y=sin(2x+φ) is shifted to the left by π/6 units along the x-axis, resulting in an even function graph. Determine the value of φ such that the equation 2(x + π/6) + φ = -x + 2πk is satisfied for some integer k. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "\\frac{\\pi}{6}",
"style": "rule"
} | {
"index": "35463",
"split": "train"
} |
math | [
{
"content": "Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's list from down to top,... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "35464",
"split": "train"
} |
math | [
{
"content": "Trapezoid $A B C D$ is inscribed in the parabola $y=x^{2}$ such that $A=\\left(a, a^{2}\\right), B=\\left(b, b^{2}\\right)$, $C=\\left(-b, b^{2}\\right)$, and $D=\\left(-a, a^{2}\\right)$ for some positive reals $a, b$ with $a>b$. If $A D+B C=A B+C D$, and $A B=\\frac{3}{4}$, what is $a$? Please o... | {
"ground_truth": "\\frac{27}{40}",
"style": "rule"
} | {
"index": "35465",
"split": "train"
} |
math | [
{
"content": "It is known that \\( b^{16} - 1 \\) has four distinct prime factors. Determine the largest one, denoted by \\( c \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "257",
"style": "rule"
} | {
"index": "35466",
"split": "train"
} |
math | [
{
"content": "Let $n$ be a positive integer. Initially, a $2n \\times 2n$ grid has $k$ black cells and the rest white cells. The following two operations are allowed :\n(1) If a $2\\times 2$ square has exactly three black cells, the fourth is changed to a black cell; \n(2) If there are exactly two black cells i... | {
"ground_truth": "n^2 + n + 1",
"style": "rule"
} | {
"index": "35467",
"split": "train"
} |
math | [
{
"content": "For any real number $x$ , we let $\\lfloor x \\rfloor$ be the unique integer $n$ such that $n \\leq x < n+1$ . For example. $\\lfloor 31.415 \\rfloor = 31$ . Compute \\[2020^{2021} - \\left\\lfloor\\frac{2020^{2021}}{2021} \\right \\rfloor (2021).\\]\n\n\n*2021 CCA Math Bonanza Team Round ... | {
"ground_truth": "2020",
"style": "rule"
} | {
"index": "35468",
"split": "train"
} |
math | [
{
"content": "A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "35469",
"split": "train"
} |
math | [
{
"content": "The points $(2, 9), (12, 14)$, and $(4, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "35470",
"split": "train"
} |
math | [
{
"content": "Given $f(\\alpha) = \\frac{\\sin(\\frac{\\pi}{2} + \\alpha) + 3\\sin(-\\pi - \\alpha)}{2\\cos(\\frac{11\\pi}{2} - \\alpha) - \\cos(5\\pi - \\alpha)}$.\n\n(I) Simplify $f(\\alpha)$;\n(II) If $\\tan \\alpha = 3$, find the value of $f(\\alpha)$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "35471",
"split": "train"
} |
math | [
{
"content": "Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. Express your answer in interval notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(0,9)",
"style": "rule"
} | {
"index": "35472",
"split": "train"
} |
math | [
{
"content": "If $a + b + c = 11$ and $ab + ac + bc = 25,$ then find\n\\[a^3 + b^3 + c^3 - 3abc.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "506",
"style": "rule"
} | {
"index": "35473",
"split": "train"
} |
math | [
{
"content": "The rational numbers $x$ and $y$, when written in lowest terms, have denominators 60 and 70 , respectively. What is the smallest possible denominator of $x+y$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "35474",
"split": "train"
} |
math | [
{
"content": "What is the greatest common divisor of $7979$ and $3713$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "79",
"style": "rule"
} | {
"index": "35475",
"split": "train"
} |
math | [
{
"content": "There are five positive integers that are common divisors of each number in the list $$36, 72, -24, 120, 96.$$ Find the sum of these five positive integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "35476",
"split": "train"
} |
math | [
{
"content": "A $\\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-d... | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "35477",
"split": "train"
} |
math | [
{
"content": "The function $f(x)$ satisfies\n\\[f(x) + 2f(1 - x) = 3x^2\\]for all real numbers $x.$ Find $f(3).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "35478",
"split": "train"
} |
math | [
{
"content": "Let $Q(x) = 0$ be the polynomial equation of the least possible degree, with rational coefficients, having $\\sqrt[4]{13} + \\sqrt[4]{169}$ as a root. Compute the product of all of the roots of $Q(x) = 0.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-13",
"style": "rule"
} | {
"index": "35479",
"split": "train"
} |
math | [
{
"content": "On each horizontal line in the figure below, the five large dots indicate the populations of cities $A, B, C, D$ and $E$ in the year indicated.\nWhich city had the greatest percentage increase in population from $1970$ to $1980$? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "C",
"style": "rule"
} | {
"index": "35480",
"split": "train"
} |
math | [
{
"content": "Given a set of seven positive integers with the unique mode being 6 and the median being 4, find the minimum possible sum of these seven integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "35481",
"split": "train"
} |
math | [
{
"content": "Three different numbers are chosen at random from the list \\(1, 3, 5, 7, 9, 11, 13, 15, 17, 19\\). The probability that one of them is the mean of the other two is \\(p\\). What is the value of \\(\\frac{120}{p}\\) ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "720",
"style": "rule"
} | {
"index": "35482",
"split": "train"
} |
math | [
{
"content": "In a regular pentagon $PQRST$, what is the measure of $\\angle PRS$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72^{\\circ}",
"style": "rule"
} | {
"index": "35483",
"split": "train"
} |
math | [
{
"content": "Circle $\\omega$ has radius 5 and is centered at $O$. Point $A$ lies outside $\\omega$ such that $OA=13$. The two tangents to $\\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\\omega$ and $\\omega$ lies out... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "35484",
"split": "train"
} |
math | [
{
"content": "For a real number $x,$ find the maximum value of\n\\[\\frac{x^4}{x^8 + 2x^6 - 4x^4 + 8x^2 + 16}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "35485",
"split": "train"
} |
math | [
{
"content": "Determine how many ordered pairs of positive integers $(x, y)$, where $x < y$, have a harmonic mean of $5^{20}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "35486",
"split": "train"
} |
math | [
{
"content": "A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "384",
"style": "rule"
} | {
"index": "35487",
"split": "train"
} |
math | [
{
"content": "The cities of Coco da Selva and Quixajuba are connected by a bus line. From Coco da Selva, buses leave for Quixajuba every hour starting at midnight. From Quixajuba, buses leave for Coco da Selva every hour starting at half past midnight. The bus journey takes exactly 5 hours.\n\nIf a bus leaves C... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "35488",
"split": "train"
} |
math | [
{
"content": "The area of the parallelogram generated by the vectors $\\mathbf{a}$ and $\\mathbf{b}$ is 8. Find the area of the parallelogram generated by the vectors $2 \\mathbf{a} + 3 \\mathbf{b}$ and $\\mathbf{a} - 5 \\mathbf{b}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "104",
"style": "rule"
} | {
"index": "35489",
"split": "train"
} |
math | [
{
"content": "A certain intelligence station has four different kinds of passwords $A$, $B$, $C$, and $D$. Each week, one of these passwords is used, and each week a password is chosen uniformly at random from the three passwords that were not used the previous week. Given that password $A$ is used in the first... | {
"ground_truth": "61/243",
"style": "rule"
} | {
"index": "35490",
"split": "train"
} |
math | [
{
"content": "What is the largest 2-digit prime factor of the integer $n = {180 \\choose 90}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "59",
"style": "rule"
} | {
"index": "35491",
"split": "train"
} |
math | [
{
"content": "How many distinct $x$-intercepts does the graph of $y = (x-5)(x^2+5x+6)$ have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "35492",
"split": "train"
} |
math | [
{
"content": "Compute: $55\\times1212-15\\times1212$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48480",
"style": "rule"
} | {
"index": "35493",
"split": "train"
} |
math | [
{
"content": "What is the coefficient of $x^8$ in the expansion of $(x-1)^9$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-9",
"style": "rule"
} | {
"index": "35494",
"split": "train"
} |
math | [
{
"content": "In an isosceles right triangle $ABC$ with $\\angle A = 90^{\\circ}$ and $AB = AC = 2$, calculate the projection of the vector $\\vec{AB}$ in the direction of $\\vec{BC}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\sqrt{2}",
"style": "rule"
} | {
"index": "35495",
"split": "train"
} |
math | [
{
"content": "During a long voyage of a passenger ship, it was observed that at each dock, a quarter of the passenger composition is renewed, that among the passengers leaving the ship, only one out of ten boarded at the previous dock, and finally, that the ship is always fully loaded.\n\nDetermine the proporti... | {
"ground_truth": "21/40",
"style": "rule"
} | {
"index": "35496",
"split": "train"
} |
math | [
{
"content": "In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 30 square units, calculate the area of trapezoid $PQRS$. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "147",
"style": "rule"
} | {
"index": "35497",
"split": "train"
} |
math | [
{
"content": "Borya and Vova play the following game on an initially white $8 \\times 8$ board. Borya goes first and, on each of his turns, colors any four white cells black. After each of his turns, Vova colors an entire row or column white. Borya aims to color as many cells black as possible, while Vova tries... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "35498",
"split": "train"
} |
math | [
{
"content": "On the ellipse $\\frac {x^{2}}{3}+ \\frac {y^{2}}{2}=1$, the distance from a point P to the left focus is $\\frac { \\sqrt {3}}{2}$. Find the distance from P to the right directrix. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": "35499",
"split": "train"
} |
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