data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "If the line $x+ay+6=0$ is parallel to the line $(a-2)x+3y+2a=0$, determine the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "32800",
"split": "train"
} |
math | [
{
"content": "A bridge is built by suspending a plank of wood between two triangular wedges with equal heights, as in the following diagram: [asy]\nimport olympiad;\nimport math;\n\n// Draw triangles\npair A = (0, 1);\npair B = (-cos(1.3962), 0);\npair C = (cos(1.3962), 0);\npair D = (2, 1);\npair E = (2-cos(1.... | {
"ground_truth": "155^\\circ",
"style": "rule"
} | {
"index": "32801",
"split": "train"
} |
math | [
{
"content": "A line with equation \\( y = 2x + b \\) passes through the point \\((-4, 0)\\). What is the value of \\(b\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "32802",
"split": "train"
} |
math | [
{
"content": "Given a rectangular box-shaped fort with dimensions 15 feet in length, 12 feet in width, and 6 feet in height, and walls and floor uniformly 1.5 feet thick, calculate the total number of one-foot cubical blocks used to build the fort. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "594",
"style": "rule"
} | {
"index": "32803",
"split": "train"
} |
math | [
{
"content": "Find the number of positive solutions to\n\\[\\sin (\\arccos( \\tan (\\arcsin x) ) ) ) = x.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "32804",
"split": "train"
} |
math | [
{
"content": "Given an ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\\frac{1}{2}$, a circle with the origin as its center and the short axis of the ellipse as its radius is tangent to the line $\\sqrt{7}x-\\sqrt{5}y+12=0$.\n(1) Find the equation of the ellipse ... | {
"ground_truth": "-\\frac{12}{7}",
"style": "rule"
} | {
"index": "32805",
"split": "train"
} |
math | [
{
"content": "A stack of logs has 15 logs on the bottom row, and each successive row has two fewer logs, ending with five special logs at the top. How many total logs are in the stack, and how many are the special logs? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "32806",
"split": "train"
} |
math | [
{
"content": "Let the three internal angles $A$, $B$, and $C$ of $\\triangle ABC$ have opposite sides $a$, $b$, and $c$ respectively, and it is given that $b(\\sin B-\\sin C)+(c-a)(\\sin A+\\sin C)=0$.\n\n$(1)$ Find the size of angle $A$;\n\n$(2)$ If $a=\\sqrt{3}$ and $\\sin C=\\frac{1+\\sqrt{3}}{2}\\sin B$, fi... | {
"ground_truth": "\\frac{3+\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "32807",
"split": "train"
} |
math | [
{
"content": "Given that $7^{-1} \\equiv 55 \\pmod{101}$, find $49^{-1} \\pmod{101}$, as a residue modulo 101 (Give an answer between 0 and 100, inclusive). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "32808",
"split": "train"
} |
math | [
{
"content": "A regular decagon $B_1B_2B_3B_4B_5B_6B_7B_8B_9B_{10}$ is inscribed in a circle of area $1$ square units. Point $Q$ lies inside the circle such that the region bounded by $\\overline{QB_1},\\overline{QB_2},$ and the minor arc $\\widehat{B_1B_2}$ of the circle has area $\\tfrac{1}{10},$ while the re... | {
"ground_truth": "110\\sqrt{3}",
"style": "rule"
} | {
"index": "32809",
"split": "train"
} |
math | [
{
"content": "Solve the equations:\n1. $2x^{2}+4x+1=0$ (using the method of completing the square)\n2. $x^{2}+6x=5$ (using the formula method) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3-\\sqrt{14}",
"style": "rule"
} | {
"index": "32810",
"split": "train"
} |
math | [
{
"content": "Each day, John ate 30% of the chocolates that were in his box at the beginning of that day. At the end of the third day, 28 chocolates remained. How many chocolates were in the box originally? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "82",
"style": "rule"
} | {
"index": "32811",
"split": "train"
} |
math | [
{
"content": "$(4^{-1} - 3^{-1})^{-1} = $ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-12",
"style": "rule"
} | {
"index": "32812",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\\overline{AB}$, and $\\overline{CD}$ bisects the right angle. The inscribed circles of $\\triangle ADC$ and $\\triangle BCD$ have radii $r_a$ and $r_b$, respectively. What is $r_a/r_b$? Please output the final answer within \\boxed{}... | {
"ground_truth": "\\frac{3}{28} \\left(10 - \\sqrt{2}\\right)",
"style": "rule"
} | {
"index": "32813",
"split": "train"
} |
math | [
{
"content": "Given a parabola \\(C\\) with the center of ellipse \\(E\\) as its focus, the parabola \\(C\\) passes through the two foci of the ellipse \\(E\\), and intersects the ellipse \\(E\\) at exactly three points. Find the eccentricity of the ellipse \\(E\\). Please output the final answer within \\boxed... | {
"ground_truth": "\\frac{2 \\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "32814",
"split": "train"
} |
math | [
{
"content": "At a robot racing competition, a certain number of mechanisms were presented. The robots were paired to race the same distance. The protocol recorded the differences in the finishing times between the winner and the loser in each of the races. All the differences were distinct: 1 sec, 2 sec, 3 sec... | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "32815",
"split": "train"
} |
math | [
{
"content": "Use the digits from $0$ to $9$ to form three-digit numbers that are divisible by $5$ and do not have repeating digits. How many such numbers are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "136",
"style": "rule"
} | {
"index": "32816",
"split": "train"
} |
math | [
{
"content": "In the triangle \\( \\triangle ABC \\), if \\( \\frac{\\overrightarrow{AB} \\cdot \\overrightarrow{BC}}{3} = \\frac{\\overrightarrow{BC} \\cdot \\overrightarrow{CA}}{2} = \\frac{\\overrightarrow{CA} \\cdot \\overrightarrow{AB}}{1} \\), find \\( \\tan A \\). Please output the final answer within \\... | {
"ground_truth": "\\sqrt{11}",
"style": "rule"
} | {
"index": "32817",
"split": "train"
} |
math | [
{
"content": "Compute the largest integer \\( k \\) such that \\( 2010^k \\) divides \\( 2010! \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "32818",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC,$ $AB = 3,$ $AC = 6,$ and $\\cos \\angle A = \\frac{1}{8}.$ Find the length of angle bisector $\\overline{AD}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "32819",
"split": "train"
} |
math | [
{
"content": "For the function $f(x) = x - 2 - \\ln x$, we know that $f(3) = 1 - \\ln 3 < 0$, $f(4) = 2 - \\ln 4 > 0$. Using the bisection method to find the approximate value of the root of $f(x)$ within the interval $(3, 4)$, we first calculate the function value $f(3.5)$. Given that $\\ln 3.5 = 1.25$, the ne... | {
"ground_truth": "3.25",
"style": "rule"
} | {
"index": "32820",
"split": "train"
} |
math | [
{
"content": "If $f(x) = \\frac{1 + x}{1 - 3x}, f_1(x) = f(f(x)), f_2(x) = f(f_1(x)),$ and in general $f_n(x) = f(f_{n-1}(x)),$ then $f_{1993}(3)=$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "32821",
"split": "train"
} |
math | [
{
"content": "Compute $i^{-100}+i^{-99}+i^{-98}+\\cdots+i^{-1}+i^0+i^1+\\cdots+i^{99}+i^{100}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "32822",
"split": "train"
} |
math | [
{
"content": "Find a six-digit number where the first digit is 6 times less than the sum of all the digits to its right, and the second digit is 6 times less than the sum of all the digits to its right. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "769999",
"style": "rule"
} | {
"index": "32823",
"split": "train"
} |
math | [
{
"content": "Given the function \\( f(x) \\) defined on \\(\\mathbf{R} \\) (the set of real numbers), it satisfies the following conditions:\n1. \\( f(1) = 1 \\)\n2. When \\( 0 < x < 1 \\), \\( f(x) > 0 \\)\n3. For any real numbers \\( x \\) and \\( y \\), the equation \\( f(x+y) - f(x-y) = 2 f(1-x) f(y) \\) h... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "32824",
"split": "train"
} |
math | [
{
"content": "In triangle \\( ABC \\) with \\( AB = 8 \\) and \\( AC = 10 \\), the incenter \\( I \\) is reflected across side \\( AB \\) to point \\( X \\) and across side \\( AC \\) to point \\( Y \\). Given that segment \\( XY \\) bisects \\( AI \\), compute \\( BC^2 \\). (The incenter \\( I \\) is the cente... | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "32825",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system, the graphs of the functions $y=\\frac{3}{x}$ and $y=x+1$ intersect at the point $\\left(m,n\\right)$. Evaluate the algebraic expression $\\left(m-n\\right)^{2}\\cdot \\left(\\frac{1}{n}-\\frac{1}{m}\\right)$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "-\\frac{1}{3}",
"style": "rule"
} | {
"index": "32826",
"split": "train"
} |
math | [
{
"content": "Let the ellipse $E: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ have an eccentricity $e = \\frac{1}{2}$, and the distance from the right focus to the line $\\frac{x}{a} + \\frac{y}{b} = 1$ be $d = \\frac{\\sqrt{21}}{7}$. Let $O$ be the origin.\n\n$(1)$ Find the equation of the ellipse $E$... | {
"ground_truth": "\\frac{2\\sqrt{21}}{7}",
"style": "rule"
} | {
"index": "32827",
"split": "train"
} |
math | [
{
"content": "Frank and Joe are playing ping pong. For each game, there is a $30 \\%$ chance that Frank wins and a $70 \\%$ chance Joe wins. During a match, they play games until someone wins a total of 21 games. What is the expected value of number of games played per match? Please output the final answer with... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "32828",
"split": "train"
} |
math | [
{
"content": "Anton, Boris, Vadim, Gena, Dima, and Egor went to the cinema. They bought 6 consecutive seats in one row. Anton and Boris want to sit next to each other, while Vadim and Gena do not want to sit next to each other. How many ways can they sit in their seats considering these preferences? Please outp... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "32829",
"split": "train"
} |
math | [
{
"content": "Given that the eccentricity of the ellipse $C:\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a > b > 0)$ is $\\frac{\\sqrt{2}}{2}$, and it passes through the point $P(\\sqrt{2},1)$. The line $y=\\frac{\\sqrt{2}}{2}x+m$ intersects the ellipse at two points $A$ and $B$.\n(1) Find the equation of the el... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "32830",
"split": "train"
} |
math | [
{
"content": "A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?\n$\\textbf{(A)}\\ \\sqrt{13}\\qquad \\textbf{(B)}\\ \\sqrt{14}\\qquad \\textbf{(C)}\\ \\sqrt... | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "32831",
"split": "train"
} |
math | [
{
"content": "Given the ellipse E: $\\\\frac{x^{2}}{4} + \\\\frac{y^{2}}{2} = 1$, O is the coordinate origin, and a line with slope k intersects ellipse E at points A and B. The midpoint of segment AB is M, and the angle between line OM and AB is θ, with tanθ = 2 √2. Find the value of k. Please output the final... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "32832",
"split": "train"
} |
math | [
{
"content": "Simplify $\\sqrt{180}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6\\sqrt5",
"style": "rule"
} | {
"index": "32833",
"split": "train"
} |
math | [
{
"content": "Ten identical crates each of dimensions $3\\mathrm{ft}\\times 4\\mathrm{ft}\\times 6\\mathrm{ft}$. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $\\fra... | {
"ground_truth": "190",
"style": "rule"
} | {
"index": "32834",
"split": "train"
} |
math | [
{
"content": "Rectangle $ABCD$ is inscribed in triangle $EFG$ such that side $AD$ of the rectangle is on side $EG$ of the triangle, as shown. The triangle's altitude from $F$ to side $EG$ is 7 inches, and $EG = 10 \\text{ inches}$. The length of segment $AB$ is equal to half the length of segment $AD$. What is ... | {
"ground_truth": " \\frac{1225}{72}",
"style": "rule"
} | {
"index": "32835",
"split": "train"
} |
math | [
{
"content": "The vertices of a $3 - 4 - 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of these circles?\n\n[asy]unitsize(1cm);\ndraw(Circle((1.8,2.4),1),linewidth(0.7));\ndraw(Circle((0,0),2),linewidth(0.7));\ndraw(Circle((5,0),3),linew... | {
"ground_truth": "14\\pi",
"style": "rule"
} | {
"index": "32836",
"split": "train"
} |
math | [
{
"content": "There are 4 male and 2 female volunteers, a total of 6 volunteers, and 2 elderly people standing in a row for a group photo. The photographer requests that the two elderly people stand next to each other in the very center, with the two female volunteers standing immediately to the left and right ... | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "32837",
"split": "train"
} |
math | [
{
"content": "In the triangle shown, $n$ is a positive integer, and $\\angle A > \\angle B > \\angle C$. How many possible values of $n$ are there? [asy]\ndraw((0,0)--(1,0)--(.4,.5)--cycle);\nlabel(\"$A$\",(.4,.5),N); label(\"$B$\",(1,0),SE); label(\"$C$\",(0,0),SW);\nlabel(\"$2n + 12$\",(.5,0),S); label(\"$3n... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "32838",
"split": "train"
} |
math | [
{
"content": "Points \\( A, B, C, D \\) are marked on a sheet of paper. A recognition device can perform two types of operations with absolute accuracy: a) measure the distance between two given points in centimeters; b) compare two given numbers. What is the minimum number of operations this device needs to pe... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "32839",
"split": "train"
} |
math | [
{
"content": "Let \\( n \\) be a natural number. Find the smallest constant \\( C \\) such that for all \\( x_{1}, \\ldots, x_{n} \\geqslant 0 \\),\n\n\\[\n\\sum_{1 \\leqslant i<j \\leqslant n} x_{i} x_{j}\\left(x_{i}^{2}+x_{j}^{2}\\right) \\leqslant C\\left(\\sum_{i=1}^{n} x_{i}\\right)^{4}.\n\\] Please output... | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "32840",
"split": "train"
} |
math | [
{
"content": "If the side of one square is the diagonal of a second square, what is the ratio of the area of the first square to the area of the second? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "32841",
"split": "train"
} |
math | [
{
"content": "The degree measure of angle $A$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "32842",
"split": "train"
} |
math | [
{
"content": "Given the function $g(x)=\\ln x+\\frac{1}{2}x^{2}-(b-1)x$.\n\n (1) If the function $g(x)$ has a monotonically decreasing interval, find the range of values for the real number $b$;\n\n (2) Let $x_{1}$ and $x_{2}$ ($x_{1} < x_{2}$) be the two extreme points of the function $g(x)$. If $b\\geqs... | {
"ground_truth": "\\frac{15}{8}-2\\ln 2",
"style": "rule"
} | {
"index": "32843",
"split": "train"
} |
math | [
{
"content": "Which of the following is equal to \\(\\frac{4}{5}\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.8",
"style": "rule"
} | {
"index": "32844",
"split": "train"
} |
math | [
{
"content": "Two years ago there were 20 trailer homes on Elm Street with an average age of 18 years. At that time, a group of brand new trailer homes was then added to Elm Street. Today, the average age of all the trailer homes on Elm Street is 14 years. How many new trailer homes were added two years ago? Pl... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "32845",
"split": "train"
} |
math | [
{
"content": "Given the parabola $C: x^{2}=2py\\left(p \\gt 0\\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\\left(y+4\\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ ... | {
"ground_truth": "20\\sqrt{5}",
"style": "rule"
} | {
"index": "32846",
"split": "train"
} |
math | [
{
"content": "Each letter represents a non-zero digit. What is the value of $t?$ \\begin{align*}\nc + o &= u \\\\\nu + n &= t \\\\\nt + c &= s \\\\\no + n + s &= 12\n\\end{align*} Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "32847",
"split": "train"
} |
math | [
{
"content": "Find the common ratio of the infinite geometric series: $$\\frac{7}{8} - \\frac{5}{12} + \\frac{35}{144} - \\dots$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{10}{21}",
"style": "rule"
} | {
"index": "32848",
"split": "train"
} |
math | [
{
"content": "At the intersection of perpendicular roads, a highway from Moscow to Kazan intersects with a road from Vladimir to Ryazan. Dima and Tolya are traveling at constant speeds from Moscow to Kazan and from Vladimir to Ryazan, respectively.\n\nWhen Dima crossed the intersection, Tolya was 900 meters awa... | {
"ground_truth": "1500",
"style": "rule"
} | {
"index": "32849",
"split": "train"
} |
math | [
{
"content": "The variables $a$ and $b$ are inversely proportional. When the sum of $a$ and $b$ is 24, their difference is 6. What is $b$ when $a$ equals 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "32850",
"split": "train"
} |
math | [
{
"content": "A solid cube of side length $4$ has $12$ edges, and a solid cube of side length $2$ has $12$ edges. The remaining solid is created by removing $8$ solid cubes of side length $2$ from the corners of the solid cube of side length $4$. Calculate the total number of edges of the remaining solid. Pleas... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "32851",
"split": "train"
} |
math | [
{
"content": "When $15$ is appended to a list of integers, the mean is increased by $2$. When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1$. How many integers were in the original list? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "32852",
"split": "train"
} |
math | [
{
"content": "$\\overline{AB}$ is a diameter of a circle. Tangents $\\overline{AD}$ and $\\overline{BC}$ are drawn so that $\\overline{AC}$ and $\\overline{BD}$ intersect in a point on the circle. If $\\overline{AD}=a$ and $\\overline{BC}=b$, $a \\not= b$, the diameter of the circle is: Please output the final ... | {
"ground_truth": "\\sqrt{ab}",
"style": "rule"
} | {
"index": "32853",
"split": "train"
} |
math | [
{
"content": "What is the remainder when $5^{207}$ is divided by 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "32854",
"split": "train"
} |
math | [
{
"content": "The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "32855",
"split": "train"
} |
math | [
{
"content": "The increasing sequence $1,3,4,9,10,12,13\\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\\mbox{th}}$ term of this sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "981",
"style": "rule"
} | {
"index": "32856",
"split": "train"
} |
math | [
{
"content": "The line RS does not intersect with △ABC. Perpendiculars are drawn from the vertices A, B, and C of △ABC to line RS, and the corresponding feet of the perpendiculars are D, E, and F respectively. It is given that AD = 10, BE = 6, and CF = 24. Let H be the foot of the perpendicular drawn from the c... | {
"ground_truth": "\\frac{40}{3}",
"style": "rule"
} | {
"index": "32857",
"split": "train"
} |
math | [
{
"content": "Let $x_1< x_2 < x_3$ be the three real roots of the equation $\\sqrt{100} x^3 - 210x^2 + 3 = 0$. Find $x_2(x_1+x_3)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "32858",
"split": "train"
} |
math | [
{
"content": "Find the minimum number of planes that divide a cube into at least 300 parts. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "32859",
"split": "train"
} |
math | [
{
"content": "Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is: Please output the final answer within \\boxed{... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "32860",
"split": "train"
} |
math | [
{
"content": "Let \\(x, y, z\\) be positive real numbers such that \\(xyz = 1\\). Find the maximum value of \n\\[\n\\frac{x^2y}{x+y} + \\frac{y^2z}{y+z} + \\frac{z^2x}{z+x}.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "32861",
"split": "train"
} |
math | [
{
"content": "Let $q(x)$ be a quadratic polynomial such that $[q(x)]^2 - x^2$ is divisible by $(x - 2)(x + 2)(x - 5)$. Find $q(10)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{110}{7}",
"style": "rule"
} | {
"index": "32862",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, angle $C$ is a right angle and the altitude from $C$ meets $\\overline{AB}$ at $D$. The lengths of the sides of $\\triangle ABC$ are integers, $BD = 17^3$, and $\\cos B = m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Please output the final answer w... | {
"ground_truth": "162",
"style": "rule"
} | {
"index": "32863",
"split": "train"
} |
math | [
{
"content": "A triangle with vertices $(6, 5)$, $(8, -3)$, and $(9, 1)$ is reflected about the line $x=8$ to create a second triangle. What is the area of the union of the two triangles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{32}{3}",
"style": "rule"
} | {
"index": "32864",
"split": "train"
} |
math | [
{
"content": "Using the digits $0$, $1$, $2$, $3$, $4$ to form a four-digit number without repeating any digit, determine the total number of four-digit numbers less than $2340$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "32865",
"split": "train"
} |
math | [
{
"content": "Let $f(x)$ be a third-degree polynomial with real coefficients satisfying \\[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\\] Find $|f(0)|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "32866",
"split": "train"
} |
math | [
{
"content": "If $x$ is a positive number such that \\[\\sqrt{8x}\\cdot\\sqrt{10x}\\cdot\\sqrt{3x}\\cdot\\sqrt{15x}=15,\\]find all possible values for $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "32867",
"split": "train"
} |
math | [
{
"content": "Given a rectangular parallelepiped with edge lengths AB = 5, BC = 3, and CG = 4, and point M as the midpoint of diagonal AG, determine the volume of the rectangular pyramid with base BCHE and apex M. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{34}",
"style": "rule"
} | {
"index": "32868",
"split": "train"
} |
math | [
{
"content": "Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3:30 PM",
"style": "rule"
} | {
"index": "32869",
"split": "train"
} |
math | [
{
"content": "Consider the hyperbola $C$: $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0)$. Its right focus and one endpoint of its conjugate axis are $F$ and $A$, respectively. Let $P$ be a point on the left branch of the hyperbola $C$. If the minimum value of the perimeter of $\\triangle APF$ is $6b$,... | {
"ground_truth": "\\frac{\\sqrt{85}}{7}",
"style": "rule"
} | {
"index": "32870",
"split": "train"
} |
math | [
{
"content": "What is the sum of all the two-digit primes that are greater than 20 but less than 80 and are still prime when their two digits are interchanged? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "291",
"style": "rule"
} | {
"index": "32871",
"split": "train"
} |
math | [
{
"content": "In any isosceles triangle $ABC$ with $AB=AC$, the altitude $AD$ bisects the base $BC$ so that $BD=DC$.\n\nDetermine the area of $\\triangle ABC$.\n\n[asy]\ndraw((0,0)--(14,0)--(7,24)--cycle,black+linewidth(1));\ndraw((7,24)--(7,0),black+linewidth(1)+dashed);\ndraw((7,0)--(7,1)--(6,1)--(6,0)--cycle... | {
"ground_truth": "168",
"style": "rule"
} | {
"index": "32872",
"split": "train"
} |
math | [
{
"content": "The center of the circle with equation $x^2+y^2=-2x-10y-16$ is the point $(x,y)$. What is $x+y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "32873",
"split": "train"
} |
math | [
{
"content": "A pawn moves on a $6 \\times 6$ chessboard. It starts from the bottom-left square. At each step, it can either jump to the square directly to its right or the square directly above it. It must reach the top-right square, such that it never lies strictly above the diagonal connecting the starting s... | {
"ground_truth": "132",
"style": "rule"
} | {
"index": "32874",
"split": "train"
} |
math | [
{
"content": "An ellipse has foci at $(9, 20)$ and $(49, 55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "32875",
"split": "train"
} |
math | [
{
"content": "How many non-congruent triangles with only integer side lengths have a perimeter of 15 units? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "32876",
"split": "train"
} |
math | [
{
"content": "Given the equation of the ellipse $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1$ with $a > b > 0$ and eccentricity $e = \\frac{\\sqrt{5}-1}{2}$, find the product of the slopes of lines $PA$ and $PB$ for a point $P$ on the ellipse that is not the left or right vertex. Please output the final answer within \... | {
"ground_truth": "\\frac{1-\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "32877",
"split": "train"
} |
math | [
{
"content": "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[f(x) f(y) - f(xy) = x + y\\]for all real numbers $x$ and $y.$\n\nLet $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \\times s.$ Please output the final answer withi... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "32878",
"split": "train"
} |
math | [
{
"content": "The diagonals of a convex quadrilateral $ABCD$, inscribed in a circle, intersect at point $E$. It is known that diagonal $BD$ is the angle bisector of $\\angle ABC$ and that $BD = 25$ and $CD = 15$. Find $BE$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15.625",
"style": "rule"
} | {
"index": "32879",
"split": "train"
} |
math | [
{
"content": "Three sectors of a circle are removed from a regular hexagon to form the shaded shape shown. Each sector has a perimeter of 18 mm. What is the perimeter, in mm, of the shaded shape formed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "32880",
"split": "train"
} |
math | [
{
"content": "Henry's little brother has $8$ identical stickers and $4$ sheets of paper, each a different color. He puts all the stickers on the pieces of paper. How many ways are there for him to do this, if only the number of stickers on each sheet of paper matters? Please output the final answer within \\box... | {
"ground_truth": "165",
"style": "rule"
} | {
"index": "32881",
"split": "train"
} |
math | [
{
"content": "Let $d$ be a number such that when 229 is divided by $d$, the remainder is 4. Compute the sum of all possible two-digit values of $d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "135",
"style": "rule"
} | {
"index": "32882",
"split": "train"
} |
math | [
{
"content": "A positive real number $x$ is such that \\[\n\\sqrt[3]{1-x^4} + \\sqrt[3]{1+x^4} = 1.\n\\]Find $x^8.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{28}{27}",
"style": "rule"
} | {
"index": "32883",
"split": "train"
} |
math | [
{
"content": "Let there be three individuals, labeled A, B, and C, to be allocated across seven laboratories in preparation for experiments. Each laboratory can accommodate no more than two people. Calculate the total number of distinct allocation schemes. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "336",
"style": "rule"
} | {
"index": "32884",
"split": "train"
} |
math | [
{
"content": "Ted's favorite number is equal to \\[1\\cdot\\binom{2007}1+2\\cdot\\binom{2007}2+3\\cdot\\binom{2007}3+\\cdots+2007\\cdot\\binom{2007}{2007}.\\] Find the remainder when Ted's favorite number is divided by $25$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "32885",
"split": "train"
} |
math | [
{
"content": "Among all the factors of $11!$ (where $11! = 11 \\times 10 \\times \\cdots \\times 1$), the largest factor that can be expressed in the form $6k + 1$ (where $k$ is a natural number) is $\\qquad$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "385",
"style": "rule"
} | {
"index": "32886",
"split": "train"
} |
math | [
{
"content": "The average of the numbers $1, 2, 3,\\dots, 98, 99,$ and $x$ is $100x$. What is $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{50}{101}",
"style": "rule"
} | {
"index": "32887",
"split": "train"
} |
math | [
{
"content": "Find the volume of a regular tetrahedron with the side of its base equal to $\\sqrt{3}$ and the angle between its lateral face and the base equal to $60^{\\circ}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "32888",
"split": "train"
} |
math | [
{
"content": "A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189$, $320$, $287$, $234$, $x$, and $y$. Find the greatest possible value of $x+y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "791",
"style": "rule"
} | {
"index": "32889",
"split": "train"
} |
math | [
{
"content": "Each vertex of a cube is to be labeled with an integer 1 through 8, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered ... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "32890",
"split": "train"
} |
math | [
{
"content": "Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 3 of them on. What is the probability that the leftmost lamp is blue and off, and the rightmost lamp is red and on? Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "\\dfrac{9}{100}",
"style": "rule"
} | {
"index": "32891",
"split": "train"
} |
math | [
{
"content": "Regular octagon CHILDREN has area 1. Determine the area of quadrilateral LINE. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "32892",
"split": "train"
} |
math | [
{
"content": "Find all 10-digit numbers \\( a_0 a_1 \\ldots a_9 \\) such that for each \\( k \\), \\( a_k \\) is the number of times that the digit \\( k \\) appears in the number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6210001000",
"style": "rule"
} | {
"index": "32893",
"split": "train"
} |
math | [
{
"content": "In the sequence $\\{a_n\\}$, if for any $n$, $a_n + a_{n+1} + a_{n+2}$ is a constant value (where $n \\in \\mathbb{N}^*$), and $a_7 = 2$, $a_9 = 3$, $a_{98} = 4$, then the sum of the first 100 terms of this sequence, $S_{100}$, equals to ______. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "299",
"style": "rule"
} | {
"index": "32894",
"split": "train"
} |
math | [
{
"content": "Let $0=x_0<x_1<\\cdots<x_n=1$ .Find the largest real number $ C$ such that for any positive integer $ n $ , we have $$ \\sum_{k=1}^n x^2_k (x_k - x_{k-1})>C $$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "32895",
"split": "train"
} |
math | [
{
"content": "A phone fully charges in 1 hour 20 minutes on fast charge and in 4 hours on regular charge. Fedya initially put the completely discharged phone on regular charge, then switched to fast charge once he found the proper adapter, completing the charging process. Determine the total charging time of th... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "32896",
"split": "train"
} |
math | [
{
"content": "On a 6 by 6 grid of points, what fraction of the larger square's area is inside the shaded square if the shaded square is rotated 45 degrees with vertices at points (2,2), (3,3), (2,4), and (1,3)? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "\\frac{1}{18}",
"style": "rule"
} | {
"index": "32897",
"split": "train"
} |
math | [
{
"content": "For each positive integer $n$, let \n\n$a_n = \\frac{(n+9)!}{(n-1)!}$.\nLet $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "32898",
"split": "train"
} |
math | [
{
"content": "A rectangle has width $x$ and length $y$. The rectangle is cut along the horizontal and vertical dotted lines to produce four smaller rectangles. The sum of the perimeters of these four rectangles is 24. What is the value of $x+y$? Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "32899",
"split": "train"
} |
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