data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "The number $x$ satisfies $5x^2 + 4 = 3x + 9$. Find the value of $(10x - 3)^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "109",
"style": "rule"
} | {
"index": "38100",
"split": "train"
} |
math | [
{
"content": "Given that the polar coordinate equation of curve C is $\\rho = \\sqrt{3}$, and the parametric equations of line l are $x = 1 + \\frac{\\sqrt{2}}{2}t$, $y = \\frac{\\sqrt{2}}{2}t$ (where t is the parameter), and the parametric equations of curve M are $x = \\cos \\theta$, $y = \\sqrt{3} \\sin \\th... | {
"ground_truth": "\\frac{3\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "38101",
"split": "train"
} |
math | [
{
"content": "Let $z$ be a complex number such that $|z - 5 - i| = 5.$ Find the minimum value of\n\\[|z - 1 + 2i|^2 + |z - 9 - 4i|^2.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "38102",
"split": "train"
} |
math | [
{
"content": "Given 18 parking spaces in a row, 14 cars arrive and occupy spaces at random, followed by Auntie Em, who requires 2 adjacent spaces, determine the probability that the remaining spaces are sufficient for her to park. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{113}{204}",
"style": "rule"
} | {
"index": "38103",
"split": "train"
} |
math | [
{
"content": "Solve the equations: \n(1) $x^2-4x+3=0$ \n(2) $4(2y-5)^2=(3y-1)^2$． Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{7}",
"style": "rule"
} | {
"index": "38104",
"split": "train"
} |
math | [
{
"content": "A play has two different male roles, two different female roles and two different roles that can be either gender. Only a man can be assigned to a male role, and only a woman can be assigned to a female role. If five men and six women audition, in how many ways can the six roles be assigned? Pleas... | {
"ground_truth": "25200",
"style": "rule"
} | {
"index": "38105",
"split": "train"
} |
math | [
{
"content": "In a modified game similar to Deal or No Deal, participants choose a box at random from a set of 30 boxes, each containing one of the following values: \n\\begin{tabular}{|c|c|}\n\\hline\n\\$0.50 & \\$2,000 \\\\\n\\hline\n\\$2 & \\$10,000 \\\\\n\\hline\n\\$10 & \\$20,000 \\\\\n\\hline\n\\$20 & \\$... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38106",
"split": "train"
} |
math | [
{
"content": "Two regular polygons have the same perimeter. If the first has 38 sides and a side length twice as long as the second, how many sides does the second have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "76",
"style": "rule"
} | {
"index": "38107",
"split": "train"
} |
math | [
{
"content": "We will call a two-digit number power-less if neither of its digits can be written as an integer to a power greater than 1. For example, 53 is power-less, but 54 is not power-less since \\(4 = 2^{2}\\). Which of the following is a common divisor of the smallest and the largest power-less numbers? ... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "38108",
"split": "train"
} |
math | [
{
"content": "Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.\n\na) How much time... | {
"ground_truth": "\\frac{29}{120}",
"style": "rule"
} | {
"index": "38109",
"split": "train"
} |
math | [
{
"content": "The numerical sequence \\(a_{0}, a_{1}, a_{2}, \\ldots \\) is such that for all non-negative \\(m\\) and \\(n\\) (where \\(m \\geq n\\)), the following relation holds:\n\n\\[a_{m+n} + a_{m-n} = \\frac{1}{2} (a_{2m} + a_{2n})\\]\n\nFind \\(a_{1995}\\) if \\(a_{1} = 1\\). Please output the final ans... | {
"ground_truth": "1995^2",
"style": "rule"
} | {
"index": "38110",
"split": "train"
} |
math | [
{
"content": "A bag contains red and yellow balls. When 60 balls are taken out, exactly 56 of them are red. Thereafter, every time 18 balls are taken out, 14 of them are always red, until the last batch of 18 balls is taken out. If the total number of red balls in the bag is exactly four-fifths of the total num... | {
"ground_truth": "336",
"style": "rule"
} | {
"index": "38111",
"split": "train"
} |
math | [
{
"content": "A clock has a hour hand $OA$ and a minute hand $OB$ with lengths of $3$ and $4$ respectively. If $0$ hour is represented as $0$ time, then the analytical expression of the area $S$ of $\\triangle OAB$ with respect to time $t$ (unit: hours) is ______, and the number of times $S$ reaches its maximum... | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "38112",
"split": "train"
} |
math | [
{
"content": "In right triangle $DEF$, we have $\\angle D = 25^\\circ$, $\\angle E = 90^\\circ$, and $EF = 9$. Find $DE$ to the nearest tenth. You may use a calculator for this problem. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19.3",
"style": "rule"
} | {
"index": "38113",
"split": "train"
} |
math | [
{
"content": "A rectangle with diagonal length $x$ is twice as long as it is wide. What is the area of the rectangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{5} x^2",
"style": "rule"
} | {
"index": "38114",
"split": "train"
} |
math | [
{
"content": "A new city is constructed in the form of 12 rectangular blocks of houses, divided by streets; at each corner of the block and at every street intersection, there is a mailbox. The blocks either touch each other along entire sides or share only a single corner.\n\nAssuming that there are 37 segment... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "38115",
"split": "train"
} |
math | [
{
"content": "In the trapezoid \\( ABCD \\) with bases \\( AD \\) and \\( BC \\), the side \\( AB \\) is equal to 2. The angle bisector of \\( \\angle BAD \\) intersects the line \\( BC \\) at point \\( E \\). A circle is inscribed in triangle \\( ABE \\), touching side \\( AB \\) at point \\( M \\) and side \\... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "38116",
"split": "train"
} |
math | [
{
"content": "A circle has a radius of 3 units. There are many line segments of length 4 units that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.\nA) $3\\pi$\nB) $5\\pi$\nC) $4\\pi$\nD) $7\\pi$\nE) $6\\pi$ Please output the final answer within \\... | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": "38117",
"split": "train"
} |
math | [
{
"content": "A number $x$ is equal to $7\\cdot24\\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "588",
"style": "rule"
} | {
"index": "38118",
"split": "train"
} |
math | [
{
"content": "A number is considered a visible factor number if it is divisible by each of its non-zero digits. For example, 204 is divisible by 2 and 4 and is therefore a visible factor number. Determine how many visible factor numbers exist from 200 to 250, inclusive. Please output the final answer within \\b... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "38119",
"split": "train"
} |
math | [
{
"content": "The ratio of the sums of the first n terms of the arithmetic sequences {a_n} and {b_n} is given by S_n / T_n = (2n) / (3n + 1). Find the ratio of the fifth terms of {a_n} and {b_n}. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{9}{14}",
"style": "rule"
} | {
"index": "38120",
"split": "train"
} |
math | [
{
"content": "The hyperbola $M$: $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ has left and right foci $F_l$ and $F_2$. The parabola $N$: $y^{2} = 2px (p > 0)$ has a focus at $F_2$. Point $P$ is an intersection point of hyperbola $M$ and parabola $N$. If the midpoint of $PF_1$ lies on the $y$... | {
"ground_truth": "\\sqrt{2} + 1",
"style": "rule"
} | {
"index": "38121",
"split": "train"
} |
math | [
{
"content": "We define two sequences of vectors $(\\mathbf{v}_n)$ and $(\\mathbf{w}_n)$ as follows: First, $\\mathbf{v}_0 = \\begin{pmatrix} 1 \\\\ 3 \\end{pmatrix},$ $\\mathbf{w}_0 = \\begin{pmatrix} 4 \\\\ 0 \\end{pmatrix}.$ Then for all $n \\ge 1,$ $\\mathbf{v}_n$ is the projection of $\\mathbf{w}_{n - 1}$... | {
"ground_truth": "\\begin{pmatrix} 4/9 \\\\ 4/3 \\end{pmatrix}",
"style": "rule"
} | {
"index": "38122",
"split": "train"
} |
math | [
{
"content": "A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "62",
"style": "rule"
} | {
"index": "38123",
"split": "train"
} |
math | [
{
"content": "Let \\(a\\) and \\(b\\) be real numbers such that\n\\[\n\\frac{a}{2b} + \\frac{a}{(2b)^2} + \\frac{a}{(2b)^3} + \\dots = 6.\n\\]\nFind\n\\[\n\\frac{a}{a + 2b} + \\frac{a}{(a + 2b)^2} + \\frac{a}{(a + 2b)^3} + \\dots.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "38124",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = x^2 + px + q$ and $g(x) = x^2 + rx + s$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g$, and the $x$-coordinate of the vertex of $g$ is a root of $f$. If both $f$ and $g$ have the same minimum value and the graphs of... | {
"ground_truth": "-200",
"style": "rule"
} | {
"index": "38125",
"split": "train"
} |
math | [
{
"content": "A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$, and that $f(x) = 1 - |x - 2|$ for $1\\leq x \\leq 3$. Find the smallest $x$ for which $f(x) = f(2001)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "429",
"style": "rule"
} | {
"index": "38126",
"split": "train"
} |
math | [
{
"content": "Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "38127",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbb{R}$ be the set of real numbers. Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a function such that for all real numbers $x$ and $y$, we have $$f\\left(x^{2}\\right)+f\\left(y^{2}\\right)=f(x+y)^{2}-2 x y$$ Let $S=\\sum_{n=-2019}^{2019} f(n)$. Determine the number of possible value... | {
"ground_truth": "2039191",
"style": "rule"
} | {
"index": "38128",
"split": "train"
} |
math | [
{
"content": "How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "58",
"style": "rule"
} | {
"index": "38129",
"split": "train"
} |
math | [
{
"content": "The sum $2\\frac{1}{7} + 3\\frac{1}{2} + 5\\frac{1}{19}$ is between Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$10\\frac{1}{2} \\text{ and } 11$",
"style": "rule"
} | {
"index": "38130",
"split": "train"
} |
math | [
{
"content": "A deck of cards now contains 54 cards, including two jokers, one being a red joker and the other a black joker, along with the standard 52 cards. In how many ways can we pick two different cards such that at least one of them is a joker? (Order matters.) Please output the final answer within \\box... | {
"ground_truth": "210",
"style": "rule"
} | {
"index": "38131",
"split": "train"
} |
math | [
{
"content": "Find the number of five-digit palindromes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "900",
"style": "rule"
} | {
"index": "38132",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be randomly selected numbers from the set $\\{1,2,3,4\\}$. Given the line $l:y=ax+b$ and the circle $O:x^{2}+y^{2}=1$, the probability that the line $l$ intersects the circle $O$ is ______. The expected number of intersection points between the line $l$ and the circle $O$ is ______... | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "38133",
"split": "train"
} |
math | [
{
"content": "How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "296",
"style": "rule"
} | {
"index": "38134",
"split": "train"
} |
math | [
{
"content": "A jacket was originally priced $\\textdollar 100$ . The price was reduced by $10\\%$ three times and increased by $10\\%$ four times in some order. To the nearest cent, what was the final price? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "106.73",
"style": "rule"
} | {
"index": "38135",
"split": "train"
} |
math | [
{
"content": "Alice needs to replace a light bulb located $10$ centimeters below the ceiling in her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the heigh... | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "38136",
"split": "train"
} |
math | [
{
"content": "Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. How many minutes did it take Joe to get from home to school? Please output the final answer within \\b... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "38137",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural number \\( n \\) such that both \\( n^2 \\) and \\( (n+1)^2 \\) contain the digit 7. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "38138",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers $n$ such that \n\\[(n - 2)(n - 4)(n - 6) \\dotsm (n - 98) > 0.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "38139",
"split": "train"
} |
math | [
{
"content": "What is the probability that Fatima gets fewer heads than tails if she flips 10 coins? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{193}{512}",
"style": "rule"
} | {
"index": "38140",
"split": "train"
} |
math | [
{
"content": "Find the value of $x$ for the following expressions.\n1. $4x^2 = 9$\n2. $(1 - 2x)^3 = 8$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "38141",
"split": "train"
} |
math | [
{
"content": "Let $z$ be a complex number. If the equation \\[x^3 + (4-i)x^2 + (2+5i)x = z\\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$ .\n\n*Proposed by Michael Tang* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "423",
"style": "rule"
} | {
"index": "38142",
"split": "train"
} |
math | [
{
"content": "For some positive integer \\( n \\), the number \\( 150n^3 \\) has \\( 150 \\) positive integer divisors, including \\( 1 \\) and the number \\( 150n^3 \\). How many positive integer divisors does the number \\( 108n^5 \\) have? Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "432",
"style": "rule"
} | {
"index": "38143",
"split": "train"
} |
math | [
{
"content": "Given an ellipse $\\frac{x^2}{25} + \\frac{y^2}{9} = 1$, the distance from a point $M$ on the ellipse to the left focus $F_1$ is 8. Find the distance from $M$ to the right directrix. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "38144",
"split": "train"
} |
math | [
{
"content": "Find the smallest digit $d$ so that $437{,}d03$ is divisible by 9. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "38145",
"split": "train"
} |
math | [
{
"content": "Given that the product of Kiana's age and the ages of her two older siblings is 256, and that they have distinct ages, determine the sum of their ages. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "38146",
"split": "train"
} |
math | [
{
"content": "What is the value of $c$ if $x\\cdot(3x+1)<c$ if and only when $x\\in \\left(-\\frac{7}{3},2\\right)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "38147",
"split": "train"
} |
math | [
{
"content": "On a luxurious ocean liner, 3000 adults consisting of men and women embark on a voyage. If 55% of the adults are men and 12% of the women as well as 15% of the men are wearing sunglasses, determine the total number of adults wearing sunglasses. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "409",
"style": "rule"
} | {
"index": "38148",
"split": "train"
} |
math | [
{
"content": "Consider an isosceles triangle $T$ with base 10 and height 12. Define a sequence $\\omega_{1}, \\omega_{2}, \\ldots$ of circles such that $\\omega_{1}$ is the incircle of $T$ and $\\omega_{i+1}$ is tangent to $\\omega_{i}$ and both legs of the isosceles triangle for $i>1$. Find the total area cont... | {
"ground_truth": "\\frac{180 \\pi}{13}",
"style": "rule"
} | {
"index": "38149",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be real numbers such that $a + 4i$ and $b + 5i$ are the roots of\n\\[z^2 - (10 + 9i) z + (4 + 46i) = 0.\\]Enter the ordered pair $(a,b).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(6,4)",
"style": "rule"
} | {
"index": "38150",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate plane $(xOy)$, if the line $ax + y - 2 = 0$ intersects the circle centered at $C$ with the equation $(x - 1)^2 + (y - a)^2 = 16$ at points $A$ and $B$, and $\\triangle ABC$ is a right triangle, then the value of the real number $a$ is _____. Please output the final answ... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "38151",
"split": "train"
} |
math | [
{
"content": "Given that \\( f(x) \\) and \\( g(x) \\) are two quadratic functions both with a leading coefficient of 1, where \\( g(6) = 35 \\) and \\( \\frac{f(-1)}{g(-1)} = \\frac{f(1)}{g(1)} = \\frac{21}{20} \\), what is \\( f(6) \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "38152",
"split": "train"
} |
math | [
{
"content": "How many ways are there to insert +'s between the digits of 111111111111111 (fifteen 1's) so that the result will be a multiple of 30? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2002",
"style": "rule"
} | {
"index": "38153",
"split": "train"
} |
math | [
{
"content": "Given are real numbers $x, y$. For any pair of real numbers $a_{0}, a_{1}$, define a sequence by $a_{n+2}=x a_{n+1}+y a_{n}$ for $n \\geq 0$. Suppose that there exists a fixed nonnegative integer $m$ such that, for every choice of $a_{0}$ and $a_{1}$, the numbers $a_{m}, a_{m+1}, a_{m+3}$, in this... | {
"ground_truth": "\\[ y = 0, 1, \\frac{1 + \\sqrt{5}}{2}, \\frac{1 - \\sqrt{5}}{2} \\]",
"style": "rule"
} | {
"index": "38154",
"split": "train"
} |
math | [
{
"content": "The first term of an arithmetic series of consecutive integers is $k^2 + 1$. The sum of $2k + 1$ terms of this series may be expressed as: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$k^3 + (k + 1)^3$",
"style": "rule"
} | {
"index": "38155",
"split": "train"
} |
math | [
{
"content": "Given that $α$ is an angle in the first quadrant, it satisfies $\\sin α - \\cos α = \\frac{\\sqrt{10}}{5}$. Find $\\cos 2α$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "- \\frac{4}{5}",
"style": "rule"
} | {
"index": "38156",
"split": "train"
} |
math | [
{
"content": "Suppose that $k \\geq 2$ is a positive integer. An in-shuffle is performed on a list with $2 k$ items to produce a new list of $2 k$ items in the following way: - The first $k$ items from the original are placed in the odd positions of the new list in the same order as they appeared in the origina... | {
"ground_truth": "83",
"style": "rule"
} | {
"index": "38157",
"split": "train"
} |
math | [
{
"content": "Given $0 \\leq a_k \\leq 1$ for $k=1,2,\\ldots,2020$, and defining $a_{2021}=a_1, a_{2022}=a_2$, find the maximum value of $\\sum_{k=1}^{2020}\\left(a_{k}-a_{k+1} a_{k+2}\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1010",
"style": "rule"
} | {
"index": "38158",
"split": "train"
} |
math | [
{
"content": "Solve the system of equations: $20=4a^{2}+9b^{2}$ and $20+12ab=(2a+3b)^{2}$. Find $ab$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{20}{3}",
"style": "rule"
} | {
"index": "38159",
"split": "train"
} |
math | [
{
"content": "Given $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, and $\\overrightarrow{a}⊥(\\overrightarrow{a}+\\overrightarrow{b})$, determine the projection of $\\overrightarrow{a}$ onto $\\overrightarrow{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "38160",
"split": "train"
} |
math | [
{
"content": "The decreasing sequence $a, b, c$ is a geometric progression, and the sequence $19a, \\frac{124b}{13}, \\frac{c}{13}$ is an arithmetic progression. Find the common ratio of the geometric progression. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "247",
"style": "rule"
} | {
"index": "38161",
"split": "train"
} |
math | [
{
"content": "Find the last three digits of $9^{105}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "049",
"style": "rule"
} | {
"index": "38162",
"split": "train"
} |
math | [
{
"content": "The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For exam... | {
"ground_truth": "761",
"style": "rule"
} | {
"index": "38163",
"split": "train"
} |
math | [
{
"content": "Two concentric circles have radii $r$ and $R>r$. Three new circles are drawn so that they are each tangent to the big two circles and tangent to the other two new circles. Find $\\frac{R}{r}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "38164",
"split": "train"
} |
math | [
{
"content": "While waiting for their next class on Killian Court, Alesha and Belinda both write the same sequence $S$ on a piece of paper, where $S$ is a 2020-term strictly increasing geometric sequence with an integer common ratio $r$. Every second, Alesha erases the two smallest terms on her paper and replac... | {
"ground_truth": "\\[\n\\boxed{2018}\n\\]",
"style": "rule"
} | {
"index": "38165",
"split": "train"
} |
math | [
{
"content": "Given that \\( a, b, c \\) are positive integers, and the parabola \\( y = ax^2 + bx + c \\) intersects the x-axis at two distinct points \\( A \\) and \\( B \\). If the distances from \\( A \\) and \\( B \\) to the origin are both less than 1, find the minimum value of \\( a + b + c \\). Please o... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "38166",
"split": "train"
} |
math | [
{
"content": "There exists \\( x_{0} < 0 \\) such that \\( x^{2} + |x - a| - 2 < 0 \\) (where \\( a \\in \\mathbb{Z} \\)) is always true. Find the sum of all values of \\( a \\) that satisfy this condition. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "38167",
"split": "train"
} |
math | [
{
"content": "Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$ .\n\n\n*2021 CCA Math Bonanza Lightning Round #3.4* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "164",
"style": "rule"
} | {
"index": "38168",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "38169",
"split": "train"
} |
math | [
{
"content": "The following image is 1024 pixels by 1024 pixels, and each pixel is either black or white. The border defines the boundaries of the image, but is not part of the image. Let $a$ be the proportion of pixels that are black. Estimate $A=\\lfloor 10000 a\\rfloor$. An estimate of $E$ will earn $\\left\... | {
"ground_truth": "3633",
"style": "rule"
} | {
"index": "38170",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive number $\\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\\in\\{z\\in C\\big| |z|<1\\}$ ,if $z_1+z_2+z_3=0$, then $$\\left|z_1z_2 +z_2z_3+z_3z_1\\right|^2+\\left|z_1z_2z_3\\right|^2 <\\lambda .$$ Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "38171",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has $AB=2 \\cdot AC$. Let $D$ and $E$ be on $\\overline{AB}$ and $\\overline{BC}$, respectively, such that $\\angle BAE = \\angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\\triangle CFE$ is equilateral. What is $\\angle ACB$? Please output the f... | {
"ground_truth": "90^\\circ",
"style": "rule"
} | {
"index": "38172",
"split": "train"
} |
math | [
{
"content": "\"Modulo $m$ graph paper\" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\\le x<m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\\equiv x^2\\pmod 5$ would cons... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38173",
"split": "train"
} |
math | [
{
"content": "Let $m$ be the product of all positive integers less than $5!$ which are invertible modulo $5!$. Find the remainder when $m$ is divided by $5!$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "119",
"style": "rule"
} | {
"index": "38174",
"split": "train"
} |
math | [
{
"content": "Let $S(n)$ denote the sum of digits of a natural number $n$ . Find all $n$ for which $n+S(n)=2004$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2001",
"style": "rule"
} | {
"index": "38175",
"split": "train"
} |
math | [
{
"content": "Let $a_{1}, a_{2}, \\ldots, a_{n}$ be a sequence of distinct positive integers such that $a_{1}+a_{2}+\\cdots+a_{n}=2021$ and $a_{1} a_{2} \\cdots a_{n}$ is maximized. If $M=a_{1} a_{2} \\cdots a_{n}$, compute the largest positive integer $k$ such that $2^{k} \\mid M$. Please output the final answ... | {
"ground_truth": "62",
"style": "rule"
} | {
"index": "38176",
"split": "train"
} |
math | [
{
"content": "Given the sequence $1990-1980+1970-1960+\\cdots -20+10$, calculate the sum. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": "38177",
"split": "train"
} |
math | [
{
"content": "Mary told John her score on the American High School Mathematics Examination (AHSME), which was over $80$. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over $80$, John could not have determined this. What was Mary... | {
"ground_truth": "119",
"style": "rule"
} | {
"index": "38178",
"split": "train"
} |
math | [
{
"content": "Determine the largest value of \\(x\\) that satisfies the equation \\(\\sqrt{3x} = 5x^2\\). Express your answer in simplest fractional form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left(\\frac{3}{25}\\right)^{\\frac{1}{3}}",
"style": "rule"
} | {
"index": "38179",
"split": "train"
} |
math | [
{
"content": "In a new diagram below, we have $\\cos \\angle XPY = \\frac{3}{5}$. A point Z is placed such that $\\angle XPZ$ is a right angle. What is $\\sin \\angle YPZ$?\n\n[asy]\npair X, P, Y, Z;\nP = (0,0);\nX = Rotate(-aCos(3/5))*(-2,0);\nY = (2,0);\nZ = Rotate(-90)*(2,0);\ndot(\"$Z$\", Z, S);\ndot(\"$Y$\... | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "38180",
"split": "train"
} |
math | [
{
"content": "How many four-digit numbers greater than 2999 can be formed such that the product of the middle two digits exceeds 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4970",
"style": "rule"
} | {
"index": "38181",
"split": "train"
} |
math | [
{
"content": "The complex number $(3 \\operatorname{cis} 18^\\circ)(-2\\operatorname{cis} 37^\\circ)$ is expressed in polar form as $r \\operatorname{cis} \\theta,$ where $r > 0$ and $0^\\circ \\le \\theta < 360^\\circ.$ Enter the ordered pair $(r, \\theta).$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "(6,235^\\circ)",
"style": "rule"
} | {
"index": "38182",
"split": "train"
} |
math | [
{
"content": "A) For a sample of size $n$ taken from a normal population with a known standard deviation $\\sigma$, the sample mean $\\bar{x}$ is found. At a significance level $\\alpha$, it is required to find the power function of the test of the null hypothesis $H_{0}: a=a_{0}$ regarding the population mean ... | {
"ground_truth": "0.8925",
"style": "rule"
} | {
"index": "38183",
"split": "train"
} |
math | [
{
"content": "Given that $a$, $b$, $c$ form an arithmetic sequence in triangle $ABC$, $\\angle B=30^{\\circ}$, and the area of $\\triangle ABC$ is $\\frac{1}{2}$, determine the value of $b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3+ \\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "38184",
"split": "train"
} |
math | [
{
"content": "Given the areas of the three squares in the figure, what is the area of the interior triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "38185",
"split": "train"
} |
math | [
{
"content": "Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a $3$-digit number with $a\\ge1$ and $a+b+c\\le7$. At the end of the trip, the odometer showed $cba$ miles. What is... | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "38186",
"split": "train"
} |
math | [
{
"content": "With all angles measured in degrees, the product $\\prod_{k=1}^{45} \\csc^2(2k-1)^\\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "91",
"style": "rule"
} | {
"index": "38187",
"split": "train"
} |
math | [
{
"content": "In $\\triangle XYZ$, the ratio $XZ:ZY$ is $5:3$. The bisector of the exterior angle at $Z$ intersects $YX$ extended at $Q$ ($Y$ is between $Q$ and $X$). Find the ratio $QY:YX$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "38188",
"split": "train"
} |
math | [
{
"content": "Calculate the value of $\\dfrac{\\sqrt[3]{81}}{\\sqrt[4]{81}}$ in terms of 81 raised to what power? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "38189",
"split": "train"
} |
math | [
{
"content": "Suppose $a$ and $b$ are positive integers for which $8 a^{a} b^{b}=27 a^{b} b^{a}$. Find $a^{2}+b^{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "117",
"style": "rule"
} | {
"index": "38190",
"split": "train"
} |
math | [
{
"content": "In the diagram below, circles \\( C_{1} \\) and \\( C_{2} \\) have centers \\( O_{1} \\) and \\( O_{2} \\), respectively. The radii of the circles are \\( r_{1} \\) and \\( r_{2} \\), with \\( r_{1} = 3r_{2} \\). Circle \\( C_{2} \\) is internally tangent to \\( C_{1} \\) at point \\( P \\). Chord... | {
"ground_truth": "160\\pi",
"style": "rule"
} | {
"index": "38191",
"split": "train"
} |
math | [
{
"content": "In quadrilateral \\(ABCD\\), \\(\\angle DAC = 98^\\circ\\), \\(\\angle DBC = 82^\\circ\\), \\(\\angle BCD = 70^\\circ\\), and \\(BC = AD\\). Find \\(\\angle ACD\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "38192",
"split": "train"
} |
math | [
{
"content": "Compute \\[\\dfrac{2^3-1}{2^3+1}\\cdot\\dfrac{3^3-1}{3^3+1}\\cdot\\dfrac{4^3-1}{4^3+1}\\cdot\\dfrac{5^3-1}{5^3+1}\\cdot\\dfrac{6^3-1}{6^3+1}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{43}{63}",
"style": "rule"
} | {
"index": "38193",
"split": "train"
} |
math | [
{
"content": "Given the equation of line $l$ is $ax+by+c=0$, where $a$, $b$, and $c$ form an arithmetic sequence, the maximum distance from the origin $O$ to the line $l$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "38194",
"split": "train"
} |
math | [
{
"content": "If the function $f(x)=\\sin \\left( \\frac{x+\\varphi}{3}\\right)$ is an even function, determine the value of $\\varphi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\pi}{2}",
"style": "rule"
} | {
"index": "38195",
"split": "train"
} |
math | [
{
"content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\\diamondsuit$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{52}",
"style": "rule"
} | {
"index": "38196",
"split": "train"
} |
math | [
{
"content": "Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid $5\\$ $. How much did the ball cost? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38197",
"split": "train"
} |
math | [
{
"content": "If $x^2+bx+9$ has two non-real roots, find all real possible values of $b$. Express your answer in interval notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-6, 6)",
"style": "rule"
} | {
"index": "38198",
"split": "train"
} |
math | [
{
"content": "Let $x_0$ be a zero of the function $f(x)=\\sin \\pi x$, and it satisfies $|x_0|+f(x_0+ \\frac{1}{2}) < 33$. Calculate the number of such zeros. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "65",
"style": "rule"
} | {
"index": "38199",
"split": "train"
} |
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