data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "How many integer solutions \\((x, y, z)\\) are there to the equation \\(xyz = 2008\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "31300",
"split": "train"
} |
math | [
{
"content": "Compute: $8 + 6(3-8)^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "158",
"style": "rule"
} | {
"index": "31301",
"split": "train"
} |
math | [
{
"content": "On a quiz, every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions on the quiz. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "31302",
"split": "train"
} |
math | [
{
"content": "Cubes. As is known, the whole space can be filled with equal cubes. At each vertex, eight cubes will converge. Therefore, by appropriately truncating the vertices of the cubes and joining the adjacent truncated parts into a single body, it is possible to fill the space with regular octahedra and t... | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "31303",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ be the intersection of the internal angle bisectors of $\\triangle ABC$. What is $BI$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "31304",
"split": "train"
} |
math | [
{
"content": "A robot colors natural numbers starting from 1 in ascending order according to the following rule: any natural number that can be expressed as the sum of two composite numbers is colored red, and those that do not meet the above criteria are colored yellow. If the numbers colored red are counted i... | {
"ground_truth": "2001",
"style": "rule"
} | {
"index": "31305",
"split": "train"
} |
math | [
{
"content": "Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy $x^{2}-y^{2}=m^{2}$ for some positive integer $m$. What is $x+y+m$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "154",
"style": "rule"
} | {
"index": "31306",
"split": "train"
} |
math | [
{
"content": "What is the value of $(2^0 - 1 + 5^2 - 0)^{-1} \\times 5?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "31307",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, $∠B≠\\frac{π}{2}$, $cos2B=\\sqrt{3}cosB-1$. \n$(1)$ Find $\\angle B$; \n$(2)$ Choose one of the three conditions, condition 1, condition 2, or condition 3, as known to ensure the existence and uniqueness of $\\triangle ABC$, and find the area of $\\triangle ABC$. \... | {
"ground_truth": "\\frac{\\sqrt{3}+2\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "31308",
"split": "train"
} |
math | [
{
"content": "Choose $3$ different numbers from the $5$ numbers $0$, $1$, $2$, $3$, $4$ to form a three-digit even number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "31309",
"split": "train"
} |
math | [
{
"content": "Six soccer teams are competing in a tournament in Waterloo. Every team is to play three games, each against a different team. How many different schedules are possible? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "31310",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression $\\log_{10} 60 + \\log_{10} 80 - \\log_{10} 15$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.505",
"style": "rule"
} | {
"index": "31311",
"split": "train"
} |
math | [
{
"content": "Keiko walks once around a track at the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Ke... | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "31312",
"split": "train"
} |
math | [
{
"content": "A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent? Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "588",
"style": "rule"
} | {
"index": "31313",
"split": "train"
} |
math | [
{
"content": "The set of points with spherical coordinates of the form\n\\[(\\rho, \\theta, \\phi) = \\left( 1, \\theta, \\frac{\\pi}{6} \\right)\\]forms a circle. Find the radius of this circle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31314",
"split": "train"
} |
math | [
{
"content": "A $2018 \\times 2018$ square was cut into rectangles with integer side lengths. Some of these rectangles were used to form a $2000 \\times 2000$ square, and the remaining rectangles were used to form a rectangle whose length differs from its width by less than 40. Find the perimeter of this rectan... | {
"ground_truth": "1076",
"style": "rule"
} | {
"index": "31315",
"split": "train"
} |
math | [
{
"content": "Points $A$, $B$, $C$, and $T$ are in space such that each of $\\overline{TA}$, $\\overline{TB}$, and $\\overline{TC}$ is perpendicular to the other two. If $TA = TB = 12$ and $TC = 6$, then what is the distance from $T$ to face $ABC$? Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "2\\sqrt{6}",
"style": "rule"
} | {
"index": "31316",
"split": "train"
} |
math | [
{
"content": "A cylinder has a radius of 3 cm and a height of 8 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31317",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1 (a > b > 0)$, the symmetric point $R$ of the left focus $F(-c,0)$ with respect to the line $bx + cy = 0$ is on the ellipse, find the eccentricity of the ellipse. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "31318",
"split": "train"
} |
math | [
{
"content": "Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the n... | {
"ground_truth": "\\frac{7}{16}",
"style": "rule"
} | {
"index": "31319",
"split": "train"
} |
math | [
{
"content": "A rectangular garden measures $12$ meters in width and $20$ meters in length. It is paved with tiles that are $2$ meters by $2$ meters each. A cat runs from one corner of the rectangular garden to the opposite corner but must leap over a small pond that exactly covers one tile in the middle of the... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "31320",
"split": "train"
} |
math | [
{
"content": "Let $x_1,$ $x_2,$ $x_3$ be positive real numbers such that $x_1 + 3x_2 + 5x_3 = 100.$ Find the smallest possible value of\n\\[ x_1^2 + x_2^2 + x_3^2. \\]\n\n--- Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2000}{7}",
"style": "rule"
} | {
"index": "31321",
"split": "train"
} |
math | [
{
"content": "Given the sequence ${a_n}$, where $a_1=1$, and $P(a_n,a_{n+1})(n∈N^{+})$ is on the line $x-y+1=0$. If the function $f(n)= \\frac {1}{n+a_{1}}+ \\frac {1}{n+a_{2}}+ \\frac {1}{n+a_{3}}+…+ \\frac {1}{n+a_{n}}(n∈N^{\\*})$, and $n\\geqslant 2$, find the minimum value of the function $f(n)$. Please out... | {
"ground_truth": "\\frac {7}{12}",
"style": "rule"
} | {
"index": "31322",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\sin \\omega x\\cos \\omega x- \\sqrt{3}\\cos^2\\omega x+ \\frac{\\sqrt{3}}{2} (\\omega > 0)$, the two adjacent axes of symmetry of its graph are $\\frac{\\pi}{2}$.\n\n$(1)$ Find the equation of the axis of symmetry for the function $y=f(x)$.\n\n$(2)$ If the zeros of the ... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "31323",
"split": "train"
} |
math | [
{
"content": "In the hexagon $ABCDEF$ with vertices $A, B, C, D, E, F$, all internal triangles dividing the hexagon are similar to isosceles triangle $ABC$, where $AB = AC$. Among these triangles, there are $10$ smallest triangles each with area $2$, and the area of triangle $ABC$ is $80$. Determine the area of... | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "31324",
"split": "train"
} |
math | [
{
"content": "Given an ellipse in the Cartesian coordinate system $xOy$, its center is at the origin, the left focus is $F(-\\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A\\left( 1,\\frac{1}{2} \\right)$.\n\n(Ⅰ) Find the standard equation of the ellipse;\n\n(Ⅱ) If a line passing through the origin... | {
"ground_truth": "\\sqrt {2}",
"style": "rule"
} | {
"index": "31325",
"split": "train"
} |
math | [
{
"content": "Let $\\overline{AB}$ be a diameter of circle $\\omega$. Extend $\\overline{AB}$ through $A$ to $C$. Point $T$ lies on $\\omega$ so that line $CT$ is tangent to $\\omega$. Point $P$ is the foot of the perpendicular from $A$ to line $CT$. Suppose $\\overline{AB} = 18$, and let $m$ denote the maximum... | {
"ground_truth": "432",
"style": "rule"
} | {
"index": "31326",
"split": "train"
} |
math | [
{
"content": "Given a cylinder of fixed volume $V,$ the total surface area (including the two circular ends) is minimized for a radius of $R$ and height $H.$ Find $\\frac{H}{R}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "31327",
"split": "train"
} |
math | [
{
"content": "A rectangular yard contains three flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, with the parallel sides measuring $10$ meters and $20$ meters. What fraction of the yard is occupied by the flower beds?\nA) $\\frac{1}{4}$\nB) $\\f... | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "31328",
"split": "train"
} |
math | [
{
"content": "In an international sprinting event, there are 10 sprinters, including 4 Americans. The medals awarded are gold, silver, and bronze. Determine the number of ways the medals can be awarded if exactly two Americans must win medals. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "216",
"style": "rule"
} | {
"index": "31329",
"split": "train"
} |
math | [
{
"content": "The graph of the function $f(x) = \\log_2 x$ is shifted 1 unit to the left, and then the part below the $x$-axis is reflected across the $x$-axis to obtain the graph of function $g(x)$. Suppose real numbers $m$ and $n$ ($m < n$) satisfy $g(m) = g\\left(-\\frac{n+1}{n+2}\\right)$ and $g(10m+6n+21) ... | {
"ground_truth": "-\\frac{1}{15}",
"style": "rule"
} | {
"index": "31330",
"split": "train"
} |
math | [
{
"content": "Given \\( f(x) = x^{5} + a_{1} x^{4} + a_{2} x^{3} + a_{3} x^{2} + a_{4} x + a_{5} \\), and \\( f(m) = 2017m \\) for \\( m = 1, 2, 3, 4 \\), calculate \\( f(10) - f(-5) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75615",
"style": "rule"
} | {
"index": "31331",
"split": "train"
} |
math | [
{
"content": "Let $n$ be a positive integer. A [i]Nordic[/i] square is an $n \\times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing... | {
"ground_truth": "2n(n - 1) + 1",
"style": "rule"
} | {
"index": "31332",
"split": "train"
} |
math | [
{
"content": "The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5}-1}{2}",
"style": "rule"
} | {
"index": "31333",
"split": "train"
} |
math | [
{
"content": "Given $\\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it satisfies $a\\cos 2C+2c\\cos A\\cos C+a+b=0$.\n\n$(1)$ Find the size of angle $C$;\n\n$(2)$ If $b=4\\sin B$, find the maximum value of the area $S$ of $\\triangle ABC$. Please ou... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "31334",
"split": "train"
} |
math | [
{
"content": "The incircle of triangle \\( ABC \\) with center \\( O \\) touches the sides \\( AB \\), \\( BC \\), and \\( AC \\) at points \\( M \\), \\( N \\), and \\( K \\) respectively. It is given that angle \\( AOC \\) is four times larger than angle \\( MKN \\). Find angle \\( B \\). Please output the fi... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "31335",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system xOy, the parametric equation of curve C is $$\\begin{cases} x=3\\cos\\theta \\\\ y=3\\sin\\theta \\end{cases}$$ (θ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the x-axis as the pola... | {
"ground_truth": "\\frac { \\sqrt {2}}{8}",
"style": "rule"
} | {
"index": "31336",
"split": "train"
} |
math | [
{
"content": "Find $p$ if $12^3=\\frac{9^2}3\\cdot2^{12p}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31337",
"split": "train"
} |
math | [
{
"content": "A standard six-sided die is rolled 3 times. If the sum of the numbers rolled on the first two rolls equals the number rolled on the third roll, what is the probability that at least one of the numbers rolled is a 2? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{8}{15}$",
"style": "rule"
} | {
"index": "31338",
"split": "train"
} |
math | [
{
"content": "As the Chinese New Year approaches, workers from a factory begin to go home to reunite with their families starting from Monday, January 17, 2011. If the number of workers leaving the factory each day is the same, and by January 31, 121 workers remain in the factory, while the total number of work... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "31339",
"split": "train"
} |
math | [
{
"content": "Three positive reals \\( x \\), \\( y \\), and \\( z \\) are such that\n\\[\n\\begin{array}{l}\nx^{2}+2(y-1)(z-1)=85 \\\\\ny^{2}+2(z-1)(x-1)=84 \\\\\nz^{2}+2(x-1)(y-1)=89\n\\end{array}\n\\]\nCompute \\( x + y + z \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "31340",
"split": "train"
} |
math | [
{
"content": "On a clock, there are two hands: the hour hand and the minute hand. At a random moment in time, the clock stops. Find the probability that the angle between the hands on the stopped clock is acute. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "31341",
"split": "train"
} |
math | [
{
"content": "Simplify the expression \\[\\sqrt{45 - 28\\sqrt{2}}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5 - 3\\sqrt{2}",
"style": "rule"
} | {
"index": "31342",
"split": "train"
} |
math | [
{
"content": "Given the equation of the parabola $y^{2}=-4x$, and the equation of the line $l$ as $2x+y-4=0$. There is a moving point $A$ on the parabola. The distance from point $A$ to the $y$-axis is $m$, and the distance from point $A$ to the line $l$ is $n$. Find the minimum value of $m+n$. Please output th... | {
"ground_truth": "\\frac{6 \\sqrt{5}}{5}-1",
"style": "rule"
} | {
"index": "31343",
"split": "train"
} |
math | [
{
"content": "In the $4 \\times 5$ grid shown, six of the $1 \\times 1$ squares are not intersected by either diagonal. When the two diagonals of an $8 \\times 10$ grid are drawn, how many of the $1 \\times 1$ squares are not intersected by either diagonal? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "31344",
"split": "train"
} |
math | [
{
"content": "Charlie folds an $\\frac{17}{2}$-inch by 11-inch piece of paper in half twice, each time along a straight line parallel to one of the paper's edges. What is the smallest possible perimeter of the piece after two such folds? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "$\\frac{39}{2}$",
"style": "rule"
} | {
"index": "31345",
"split": "train"
} |
math | [
{
"content": "The set of points $(x,y)$ such that $|x - 3| \\le y \\le 4 - |x - 1|$ defines a region in the $xy$-plane. Compute the area of this region. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "31346",
"split": "train"
} |
math | [
{
"content": "In a mathematics competition consisting of three problems, A, B, and C, among the 39 participants, each person solved at least one problem. Among those who solved problem A, there are 5 more people who only solved A than those who solved A and any other problems. Among those who did not solve prob... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "31347",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides have integer lengths and $AB=AC$. Circle $\\omega$ has its center at the incenter of $\\triangle ABC$. An excircle of $\\triangle ABC$ is a circle in the exterior of $\\triangle ABC$ that is tangent to one side of the triangle and tangent to the extensions of the oth... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "31348",
"split": "train"
} |
math | [
{
"content": "Monica is tiling the floor of her 12-foot by 16-foot living room. She plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will she use? Please output the final answ... | {
"ground_truth": "87",
"style": "rule"
} | {
"index": "31349",
"split": "train"
} |
math | [
{
"content": "Each of $b_1, b_2, \\dots, b_{150}$ is equal to $2$ or $-2$. Find the minimum positive value of\n\\[\\sum_{1 \\le i < j \\le 150} b_i b_j.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "31350",
"split": "train"
} |
math | [
{
"content": "What integer \\( n \\) satisfies \\( 0 \\le n < 23 \\) and\n$$\n54126 \\equiv n \\pmod{23}~?\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "31351",
"split": "train"
} |
math | [
{
"content": "Given the data in the table, where the number of No. 5 batteries is $x$ and the number of No. 7 batteries is $y$, the masses of one No. 5 battery and one No. 7 battery are $x$ grams and $y$ grams, respectively. By setting up and solving a system of equations, express the value of $x$ obtained by e... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "31352",
"split": "train"
} |
math | [
{
"content": "In the diagram, each circle is divided into two equal areas and $O$ is the center of the larger circle. The area of the larger circle is $64\\pi.$ What is the total area of the shaded regions? [asy]\nsize(100);\nimport graph;\nfill(Arc((0,0),2,180,360)--cycle,mediumgray);fill(Arc((0,1),1,0,180)--c... | {
"ground_truth": "40\\pi",
"style": "rule"
} | {
"index": "31353",
"split": "train"
} |
math | [
{
"content": "The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first $30,000$ miles the car traveled. For how many miles was each tire used? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24000",
"style": "rule"
} | {
"index": "31354",
"split": "train"
} |
math | [
{
"content": "If \\( x_{i}=\\frac{i}{101} \\), then the value of \\( S=\\sum_{i=0}^{101} \\frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \\) is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "31355",
"split": "train"
} |
math | [
{
"content": "Let \\( O \\) be a point inside \\( \\triangle ABC \\), such that \\(\\overrightarrow{AO} = \\frac{1}{3}\\overrightarrow{AB} + \\frac{1}{4}\\overrightarrow{AC}\\). Find the ratio of the area of \\( \\triangle OAB \\) to the area of \\( \\triangle OBC \\). Please output the final answer within \\bo... | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "31356",
"split": "train"
} |
math | [
{
"content": "Let \\( m \\) be an integer greater than 1, and let's define a sequence \\( \\{a_{n}\\} \\) as follows:\n\\[\n\\begin{array}{l}\na_{0}=m, \\\\\na_{1}=\\varphi(m), \\\\\na_{2}=\\varphi^{(2)}(m)=\\varphi(\\varphi(m)), \\\\\n\\vdots \\\\\na_{n}=\\varphi^{(n)}(m)=\\varphi\\left(\\varphi^{(n-1)}(m)\\ri... | {
"ground_truth": "1944",
"style": "rule"
} | {
"index": "31357",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}=(1,\\sin x)$, $\\overrightarrow{b}=(\\sin x,-1)$, $\\overrightarrow{c}=(1,\\cos x)$, where $x\\in(0,\\pi)$. \n(Ⅰ) If $(\\overrightarrow{a}+ \\overrightarrow{b})\\nparallel \\overrightarrow{c}$, find $x$; \n(Ⅱ) In $\\triangle ABC$, the sides opposite to angles $... | {
"ground_truth": "\\frac{1-3\\sqrt{5}}{8}",
"style": "rule"
} | {
"index": "31358",
"split": "train"
} |
math | [
{
"content": "The sequences of positive integers $1,a_2, a_3, \\dots$ and $1,b_2, b_3, \\dots$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$. Please output the final answ... | {
"ground_truth": "262",
"style": "rule"
} | {
"index": "31359",
"split": "train"
} |
math | [
{
"content": "Three congruent cones, each with a radius of 8 cm and a height of 8 cm, are enclosed within a cylinder. The base of each cone is consecutively stacked and forms a part of the cylinder’s interior base, while the height of the cylinder is 24 cm. Calculate the volume of the cylinder that is not occup... | {
"ground_truth": "1024\\pi",
"style": "rule"
} | {
"index": "31360",
"split": "train"
} |
math | [
{
"content": "Given the two functions $f(x)=x^2+1$ and $g(x)=2x-1$, find $f(g(5))$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "82",
"style": "rule"
} | {
"index": "31361",
"split": "train"
} |
math | [
{
"content": "A bot is held in orbit by the gravitational force of attraction to the planet, which creates a centripetal acceleration equal to $4 \\pi^{2}(2 R) / T^{2}$, where $R$ is the radius of the planet and $T$ is the period of revolution of the bot. From Newton's second law, we have:\n\n$$\nm \\frac{4 \\p... | {
"ground_truth": "6000",
"style": "rule"
} | {
"index": "31362",
"split": "train"
} |
math | [
{
"content": "Given that $\\theta$ is an angle in the second quadrant, and $\\tan(\\theta + \\frac{\\pi}{4}) = \\frac{1}{2}$, calculate the value of $\\sin\\theta + \\cos\\theta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\sqrt{10}}{5}",
"style": "rule"
} | {
"index": "31363",
"split": "train"
} |
math | [
{
"content": "Given that the terminal side of angle $θ$ lies on the ray $y=2x(x≤0)$, find the value of $\\sin θ + \\cos θ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "31364",
"split": "train"
} |
math | [
{
"content": "A number composed of ten million, three hundred thousand, and fifty is written as \\_\\_\\_\\_\\_\\_, and this number is read as \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10300050",
"style": "rule"
} | {
"index": "31365",
"split": "train"
} |
math | [
{
"content": "The day before yesterday, students collected 3 kg more scrap paper than they did yesterday, and yesterday they collected 40 kg less than the total amount collected the day before yesterday and today combined. How many kilograms of scrap paper did the students collect today? Please output the final... | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "31366",
"split": "train"
} |
math | [
{
"content": "If $\\sqrt{n+9}=25$, what is the value of $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "616",
"style": "rule"
} | {
"index": "31367",
"split": "train"
} |
math | [
{
"content": "Evaluate the product \\[ (a-10) \\cdot (a-9) \\cdot \\dotsm \\cdot (a-1) \\cdot a, \\] where $a=2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "31368",
"split": "train"
} |
math | [
{
"content": "From a bottle containing 1 liter of alcohol, $\\frac{1}{3}$ liter of alcohol is poured out, an equal amount of water is added and mixed thoroughly. Then, $\\frac{1}{3}$ liter of the mixture is poured out, an equal amount of water is added and mixed thoroughly. Finally, 1 liter of the mixture is po... | {
"ground_truth": "\\frac{8}{27}",
"style": "rule"
} | {
"index": "31369",
"split": "train"
} |
math | [
{
"content": "For what real values of $a$ is the expression $\\frac{a+3}{a^2-4}$ undefined? List your answers in increasing order separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2, 2",
"style": "rule"
} | {
"index": "31370",
"split": "train"
} |
math | [
{
"content": "Find the sum of all positive integers $n \\leq 2015$ that can be expressed in the form $\\left\\lceil\\frac{x}{2}\\right\\rceil+y+x y$, where $x$ and $y$ are positive integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2029906",
"style": "rule"
} | {
"index": "31371",
"split": "train"
} |
math | [
{
"content": "During the New Year's Day holiday of 2018, 8 high school students from four different classes, each with 2 students, plan to carpool for a trip. Classes are denoted as (1), (2), (3), and (4). They will be divided between two cars, Car A and Car B, each with a capacity for 4 students (seating posit... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "31372",
"split": "train"
} |
math | [
{
"content": "The angle of inclination of the line $x-y-1=0$ is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{\\pi}{4}",
"style": "rule"
} | {
"index": "31373",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\cos (\\omega x+\\varphi)$ ($\\omega > 0$, $|\\varphi| \\leqslant \\frac {\\pi}{2}$), when $x=- \\frac {\\pi}{4}$, the function $f(x)$ can achieve its minimum value, and when $x= \\frac {\\pi}{4}$, the function $y=f(x)$ can achieve its maximum value. Moreover, $f(x)$ is m... | {
"ground_truth": "- \\frac {\\pi}{2}",
"style": "rule"
} | {
"index": "31374",
"split": "train"
} |
math | [
{
"content": "Find the number of ways in which the letters in \"HMMTHMMT\" can be rearranged so that each letter is adjacent to another copy of the same letter. For example, \"MMMMTTHH\" satisfies this property, but \"HHTMMMTM\" does not. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "31375",
"split": "train"
} |
math | [
{
"content": "Given the numbers $1$, $2$, $3$, $4$, $5$, randomly select $3$ numbers (with repetition allowed) to form a three-digit number, find the probability that the sum of its digits equals $12$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{2}{25}",
"style": "rule"
} | {
"index": "31376",
"split": "train"
} |
math | [
{
"content": "A piece of alloy weighing 6 kg contains copper. Another piece of alloy weighing 8 kg contains copper in a different percentage than the first piece. A certain part was separated from the first piece, and a part twice as heavy was separated from the second piece. Each of the separated parts was the... | {
"ground_truth": "2.4",
"style": "rule"
} | {
"index": "31377",
"split": "train"
} |
math | [
{
"content": "How many integers between 100 and 300 have both 11 and 8 as factors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "31378",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\lfloor{\\sqrt{12}}\\rfloor^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "31379",
"split": "train"
} |
math | [
{
"content": "A right triangle has area 5 and a hypotenuse of length 5. Find its perimeter. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5+3 \\sqrt{5}",
"style": "rule"
} | {
"index": "31380",
"split": "train"
} |
math | [
{
"content": "Find all $t$ such that $x-t$ is a factor of $4x^2 + 11x - 3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "31381",
"split": "train"
} |
math | [
{
"content": "Péter sent his son with a message to his brother, Károly, who in turn sent his son to Péter with a message. The cousins met $720 \\, \\text{m}$ away from Péter's house, had a 2-minute conversation, and then continued on their way. Both boys spent 10 minutes at the respective relative's house. On t... | {
"ground_truth": "1760",
"style": "rule"
} | {
"index": "31382",
"split": "train"
} |
math | [
{
"content": "A conference hall is setting up seating for a workshop. Each row must contain $13$ chairs, and initially, there are $169$ chairs in total. The organizers expect $95$ participants to attend the workshop. To ensure all rows are completely filled with minimal empty seats, how many chairs should be re... | {
"ground_truth": "65",
"style": "rule"
} | {
"index": "31383",
"split": "train"
} |
math | [
{
"content": "At the start of this month, Mathilde and Salah each had 100 coins. For Mathilde, this was 25% more coins than she had at the start of last month. For Salah, this was 20% fewer coins than he had at the start of last month. What was the total number of coins that they had at the start of last month?... | {
"ground_truth": "205",
"style": "rule"
} | {
"index": "31384",
"split": "train"
} |
math | [
{
"content": "Given the regression equation $y=0.75x-68.2$ and a student's height of $x=170$ cm, calculate the student's weight in kg. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "59.3",
"style": "rule"
} | {
"index": "31385",
"split": "train"
} |
math | [
{
"content": "Three $\\Delta$'s and a $\\diamondsuit$ will balance nine $\\bullet$'s. One $\\Delta$ will balance a $\\diamondsuit$ and a $\\bullet$.\n\nHow many $\\bullet$'s will balance the two $\\diamondsuit$'s in this balance? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "31386",
"split": "train"
} |
math | [
{
"content": "Compute: $\\frac{1}{5} + \\frac{2}{5} + \\frac{3}{5} + \\dots + \\frac{9}{5} + \\frac{10}{5}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31387",
"split": "train"
} |
math | [
{
"content": "In a given area, there are 10 famous tourist attractions, of which 8 are for daytime visits and 2 are for nighttime visits. A tour group wants to select 5 from these 10 spots for a two-day tour. The itinerary is arranged with one spot in the morning, one in the afternoon, and one in the evening of... | {
"ground_truth": "2352",
"style": "rule"
} | {
"index": "31388",
"split": "train"
} |
math | [
{
"content": "Real numbers $x$ and $y$ satisfy\n\\begin{align*}\nx^2 + y^2 &= 2023 \n(x-2)(y-2) &= 3.\n\\end{align*}\nFind the largest possible value of $|x-y|$ .\n\n*Proposed by Howard Halim* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13\\sqrt{13}",
"style": "rule"
} | {
"index": "31389",
"split": "train"
} |
math | [
{
"content": "Given that $\\frac{\\cos 2\\alpha}{\\sqrt{2}\\sin\\left(\\alpha+\\frac{\\pi}{4}\\right)}=\\frac{\\sqrt{5}}{2}$, find the value of $\\tan\\alpha+\\frac{1}{\\tan\\alpha}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "31390",
"split": "train"
} |
math | [
{
"content": "Let $\\Omega$ be a sphere of radius 4 and $\\Gamma$ be a sphere of radius 2 . Suppose that the center of $\\Gamma$ lies on the surface of $\\Omega$. The intersection of the surfaces of $\\Omega$ and $\\Gamma$ is a circle. Compute this circle's circumference. Please output the final answer within \... | {
"ground_truth": "\\pi \\sqrt{15}",
"style": "rule"
} | {
"index": "31391",
"split": "train"
} |
math | [
{
"content": "In the $xy$-plane, consider a T-shaped region bounded by horizontal and vertical segments with vertices at $(0,0), (0,4), (4,4), (4,2), (7,2), (7,0)$. What is the slope of the line through the origin that divides the area of this region exactly in half?\nA) $\\frac{1}{4}$\nB) $\\frac{1}{3}$\nC) $\... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31392",
"split": "train"
} |
math | [
{
"content": "Find all positive integers $n$ such that the product of all positive divisors of $n$ is $24^{240}$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24^5",
"style": "rule"
} | {
"index": "31393",
"split": "train"
} |
math | [
{
"content": "A regular hexagon $A B C D E F$ has side length 1 and center $O$. Parabolas $P_{1}, P_{2}, \\ldots, P_{6}$ are constructed with common focus $O$ and directrices $A B, B C, C D, D E, E F, F A$ respectively. Let $\\chi$ be the set of all distinct points on the plane that lie on at least two of the s... | {
"ground_truth": "35 \\sqrt{3}",
"style": "rule"
} | {
"index": "31394",
"split": "train"
} |
math | [
{
"content": "The radius of the circumcircle of the acute-angled triangle \\(ABC\\) is 1. It is known that on this circumcircle lies the center of another circle passing through the vertices \\(A\\), \\(C\\), and the orthocenter of triangle \\(ABC\\). Find \\(AC\\). Please output the final answer within \\boxed... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "31395",
"split": "train"
} |
math | [
{
"content": "Take a clay sphere of radius 13, and drill a circular hole of radius 5 through its center. Take the remaining \"bead\" and mold it into a new sphere. What is this sphere's radius? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "31396",
"split": "train"
} |
math | [
{
"content": "A box contains six cards. Three of the cards are black on both sides, one card is black on one side and red on the other, and two of the cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probab... | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "31397",
"split": "train"
} |
math | [
{
"content": "Let \\[ f(x) =\n\\begin{cases}\n-x^2 & \\text{if } x \\geq 0,\\\\\nx+8& \\text{if } x <0.\n\\end{cases}\n\\]Compute $f(f(f(f(f(1))))).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-33",
"style": "rule"
} | {
"index": "31398",
"split": "train"
} |
math | [
{
"content": "Let $m, n, p$ be fixed positive real numbers which satisfy $mnp = 8$. Depending on these constants, find the minimum of $$x^2+y^2+z^2+ mxy + nxz + pyz,$$\nwhere $x, y, z$ are arbitrary positive real numbers satisfying $xyz = 8$. When is the equality attained?\nSolve the problem for:\n[list=a][*]$m... | {
"ground_truth": "z=\\sqrt[3]{4m}",
"style": "rule"
} | {
"index": "31399",
"split": "train"
} |
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