data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "A right triangular pyramid has a base edge length of $2$, and its three side edges are pairwise perpendicular. Calculate the volume of this pyramid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{6}}{3}",
"style": "rule"
} | {
"index": "1200",
"split": "train"
} |
math | [
{
"content": "For any positive integer $n$, define $\\boxed{n}$ to be the sum of the positive factors of $n$.\nFor example, $\\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\\boxed{\\boxed{11}}$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "1201",
"split": "train"
} |
math | [
{
"content": "Let $x, y, a \\in \\mathbb{R}^*$, and when $x + 2y = 1$, the minimum value of $\\frac{3}{x} + \\frac{a}{y}$ is $6\\sqrt{3}$. Determine the minimum value of $3x + ay$ when $\\frac{1}{x} + \\frac{2}{y} = 1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6\\sqrt{3}",
"style": "rule"
} | {
"index": "1202",
"split": "train"
} |
math | [
{
"content": "The number of distinct pairs of integers $(x, y)$ such that $0<x<y$ and $\\sqrt{1984}=\\sqrt{x}+\\sqrt{y}$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "1203",
"split": "train"
} |
math | [
{
"content": "Let\n\\[x^5 + x^3 - x^2 - 1 = q_1(x) q_2(x) \\dotsm q_m(x),\\]\nwhere each non-constant polynomial $q_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers. Compute $q_1(2) + q_2(2) + \\dots + q_m(2).$ Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "1204",
"split": "train"
} |
math | [
{
"content": "The sum of the numbers 300, 2020, and 10001 is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12321",
"style": "rule"
} | {
"index": "1205",
"split": "train"
} |
math | [
{
"content": "A rental company owns 100 cars. When the monthly rent for each car is 3000 yuan, all of them can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan per month, and for each ca... | {
"ground_truth": "307050",
"style": "rule"
} | {
"index": "1206",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \\[\\sin^2x_1+\\sin^2x_2+\\sin^2x_3+\\cdots+\\sin^2x_{2007},\\] where, for $1\\leq i\\leq 2007$ , $x_i$ is a nonnegative real number, and \\[x_1+x_2+x_3+\\cdots+x_{2007}=\\pi.\\] Find th... | {
"ground_truth": "2008",
"style": "rule"
} | {
"index": "1207",
"split": "train"
} |
math | [
{
"content": "Given a regular polygon with $n$ sides. It is known that there are $1200$ ways to choose three of the vertices of the polygon such that they form the vertices of a **right triangle**. What is the value of $n$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "1208",
"split": "train"
} |
math | [
{
"content": "Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "198",
"style": "rule"
} | {
"index": "1209",
"split": "train"
} |
math | [
{
"content": "Suppose you have 3 red candies, 2 green candies, and 4 blue candies. A flavor is defined by the ratio of red to green to blue candies, and different flavors are identified by distinct ratios (flavors are considered the same if they can be reduced to the same ratio). How many different flavors can ... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "1210",
"split": "train"
} |
math | [
{
"content": "27 identical dice were glued together to form a $3 \\times 3 \\times 3$ cube in such a way that any two adjacent small dice have the same number of dots on the touching faces. How many dots are there on the surface of the large cube? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "189",
"style": "rule"
} | {
"index": "1211",
"split": "train"
} |
math | [
{
"content": "The functions $f(x) = x^2-2x + m$ and $g(x) = x^2-2x + 4m$ are evaluated when $x = 4$. What is the value of $m$ if $2f(4) = g(4)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "1212",
"split": "train"
} |
math | [
{
"content": "How many integers $-15 \\leq n \\leq 10$ satisfy $(n-1)(n+3)(n + 7) < 0$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "1213",
"split": "train"
} |
math | [
{
"content": "There are 7 people standing in a row. How many different arrangements are there according to the following requirements?\n(1) Among them, A, B, and C cannot stand next to each other;\n(2) Among them, A and B have exactly one person between them;\n(3) A does not stand at the head of the row, and B ... | {
"ground_truth": "3720",
"style": "rule"
} | {
"index": "1214",
"split": "train"
} |
math | [
{
"content": "In $\\triangle PQR$, point $T$ is on side $QR$ such that $QT=6$ and $TR=10$. What is the ratio of the area of $\\triangle PQT$ to the area of $\\triangle PTR$? [asy]\nsize(6cm);\npair q = (0, 0); pair t = (6, 0); pair r = (16, 0);\npair p = (4, 8);\ndraw(p--q--r--cycle--t);\nlabel(\"$P$\", p, N);\... | {
"ground_truth": "3:5",
"style": "rule"
} | {
"index": "1215",
"split": "train"
} |
math | [
{
"content": "A team of loggers was scheduled to harvest $216 \\mathrm{~m}^{3}$ of wood over several days. For the first three days, the team met the daily target set by the plan. Then, they harvested an additional $8 \\mathrm{~m}^{3}$ above the daily target each day. As a result, they harvested $232 \\mathrm{~... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "1216",
"split": "train"
} |
math | [
{
"content": "Let \\( a, b, c, d \\) be integers such that \\( a > b > c > d \\geq -2021 \\) and\n\\[ \\frac{a+b}{b+c} = \\frac{c+d}{d+a} \\]\n(and \\( b+c \\neq 0 \\neq d+a \\)). What is the maximum possible value of \\( a \\cdot c \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "510050",
"style": "rule"
} | {
"index": "1217",
"split": "train"
} |
math | [
{
"content": "Four people are sitting around a round table, with identical coins placed in front of each person. Everyone flips their coin simultaneously. If the coin lands heads up, the person stands up; if it lands tails up, the person remains seated. Calculate the probability that no two adjacent people stan... | {
"ground_truth": "\\frac{7}{16}",
"style": "rule"
} | {
"index": "1218",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{p}$ be the projection of vector $\\mathbf{v}$ onto vector $\\mathbf{u},$ and let $\\mathbf{q}$ be the projection of $\\mathbf{p}$ onto $\\mathbf{u}.$ If $\\frac{\\|\\mathbf{p}\\|}{\\|\\mathbf{v}\\|} = \\frac{3}{4},$ then find $\\frac{\\|\\mathbf{q}\\|}{\\|\\mathbf{u}\\|}.$ Please out... | {
"ground_truth": "\\frac{9}{16}",
"style": "rule"
} | {
"index": "1219",
"split": "train"
} |
math | [
{
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and there exists a point $P$ on the hyperbola such that $\\angle F\\_1PF\\_2 = 60^{\\circ}$ and $|OP| = 3b$ (where $O$ is the origin), find the eccentricity of the ... | {
"ground_truth": "\\frac{\\sqrt{42}}{6}",
"style": "rule"
} | {
"index": "1220",
"split": "train"
} |
math | [
{
"content": "In the diagram below, $WXYZ$ is a trapezoid such that $\\overline{WX}\\parallel \\overline{ZY}$ and $\\overline{WY}\\perp\\overline{ZY}$. If $YZ = 15$, $\\tan Z = \\frac{4}{3}$, and $\\tan X = \\frac{3}{2}$, what is the length of $XY$? Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "\\frac{20\\sqrt{13}}{3}",
"style": "rule"
} | {
"index": "1221",
"split": "train"
} |
math | [
{
"content": "Roll a die twice. Let $X$ be the maximum of the two numbers rolled. Which of the following numbers is closest to the expected value $E(X)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "1222",
"split": "train"
} |
math | [
{
"content": "Let p, q, r, s, and t be distinct integers such that (9-p)(9-q)(9-r)(9-s)(9-t) = -120. Calculate the value of p+q+r+s+t. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "1223",
"split": "train"
} |
math | [
{
"content": "Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$, and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\\triangle GEM$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "1224",
"split": "train"
} |
math | [
{
"content": "Simplify $2y + 3y+4y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9y",
"style": "rule"
} | {
"index": "1225",
"split": "train"
} |
math | [
{
"content": "In a right trapezoid \\(ABCD\\), the sum of the lengths of the bases \\(AD\\) and \\(BC\\) is equal to its height \\(AB\\). In what ratio does the angle bisector of \\(\\angle ABC\\) divide the lateral side \\(CD\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1:1",
"style": "rule"
} | {
"index": "1226",
"split": "train"
} |
math | [
{
"content": "Calculate the product of $1101_2 \\cdot 111_2$. Express your answer in base 2. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1000111_2",
"style": "rule"
} | {
"index": "1227",
"split": "train"
} |
math | [
{
"content": "The area of the base of a hemisphere is $144\\pi$. A cylinder of the same radius as the hemisphere and height equal to the radius of the hemisphere is attached to its base. What is the total surface area of the combined solid (hemisphere + cylinder)? Please output the final answer within \\boxed{}... | {
"ground_truth": "576\\pi",
"style": "rule"
} | {
"index": "1228",
"split": "train"
} |
math | [
{
"content": "If $x + y = 16$ and $x-y = 2$, what is the value of $x^2 - y^2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "1229",
"split": "train"
} |
math | [
{
"content": "$ABCDEFGH$ shown below is a cube. Find $\\sin \\angle HAC$.\n\n[asy]\n\nimport three;\n\ntriple A,B,C,D,EE,F,G,H;\n\nA = (0,0,0);\n\nB = (1,0,0);\n\nC = (1,1,0);\n\nD= (0,1,0);\n\nEE = (0,0,1);\n\nF = B+EE;\n\nG = C + EE;\n\nH = D + EE;\n\ndraw(B--C--D);\n\ndraw(B--A--D,dashed);\n\ndraw(EE--F--G--... | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "1230",
"split": "train"
} |
math | [
{
"content": "Given that $k$ is a composite number, and $1 < k < 100$, when the sum of the digits of $k$ is a prime number, this prime number is called the \"derived prime\" of $k$. \n(1) If the \"derived prime\" of $k$ is 2, then $k=$ \n(2) Let set $A=\\{P(k) | P(k)$ is the \"derived prime\" of $k\\}$, $B=\\... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "1231",
"split": "train"
} |
math | [
{
"content": "Simplify and evaluate:\n(1) Calculate the value of $\\frac {1}{\\log_{4}6}+6^{\\log_{6} \\sqrt {3}-1}-2\\log_{6} \\frac {1}{3}$;\n(2) Given $\\tan\\alpha=2$ and $\\sin\\alpha+\\cos\\alpha < 0$, find the value of $\\frac {\\tan(\\pi-\\alpha)\\cdot \\sin(-\\alpha+ \\frac {3\\pi}{2})}{\\cos(\\pi +\\a... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "1232",
"split": "train"
} |
math | [
{
"content": "Select 3 people from 3 boys and 2 girls to participate in a speech competition. \n(1) Calculate the probability that the 3 selected people are all boys; \n(2) Calculate the probability that exactly 1 of the 3 selected people is a girl; \n(3) Calculate the probability that at least 1 of the 3 se... | {
"ground_truth": "\\dfrac{9}{10}",
"style": "rule"
} | {
"index": "1233",
"split": "train"
} |
math | [
{
"content": "Right triangle $ABC$ has its right angle at $A$ . A semicircle with center $O$ is inscribed inside triangle $ABC$ with the diameter along $AB$ . Let $D$ be the point where the semicircle is tangent to $BC$ . If $AD=4$ and $CO=5$ , find $\\cos\\angle{ABC}$ .\n\n[asy]\nimport olympiad... | {
"ground_truth": "$\\frac{4}{5}$",
"style": "rule"
} | {
"index": "1234",
"split": "train"
} |
math | [
{
"content": "Let $N = 34 \\cdot 34 \\cdot 63 \\cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 : 14",
"style": "rule"
} | {
"index": "1235",
"split": "train"
} |
math | [
{
"content": "Mrs. Delta's language class has 52 students, each with unique initials, and no two students have initials that are alphabetically consecutive (e.g., AB cannot follow AC directly). Assuming Y is considered a consonant, what is the probability of randomly picking a student whose initials (each first... | {
"ground_truth": "\\frac{5}{52}",
"style": "rule"
} | {
"index": "1236",
"split": "train"
} |
math | [
{
"content": "Let the common ratio of the geometric sequence $\\{a_n\\}$ be $q$, and the sum of the first $n$ terms be $S_n$. If $S_{n+1}$, $S_n$, and $S_{n+2}$ form an arithmetic sequence, find the value of $q$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "1237",
"split": "train"
} |
math | [
{
"content": "The lengths of the diagonals of a rhombus and the length of its side form a geometric progression. Find the sine of the angle between the side of the rhombus and its longer diagonal, given that it is greater than \\( \\frac{1}{2} \\). Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "\\sqrt{\\frac{\\sqrt{17}-1}{8}}",
"style": "rule"
} | {
"index": "1238",
"split": "train"
} |
math | [
{
"content": "The fraction $\\frac{2(\\sqrt2+\\sqrt6)}{3\\sqrt{2+\\sqrt3}}$ is equal to Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac43",
"style": "rule"
} | {
"index": "1239",
"split": "train"
} |
math | [
{
"content": "Let $A = (-4, 0),$ $B=(-3,2),$ $C=(3,2),$ and $D=(4,0).$ Suppose that point $P$ satisfies \\[PA + PD = PB + PC = 10.\\]Find the $y-$coordinate of $P,$ and express it in its simplest form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{6}{7}",
"style": "rule"
} | {
"index": "1240",
"split": "train"
} |
math | [
{
"content": "The digits of a four-digit positive integer add up to 14. The sum of the two middle digits is nine, and the thousands digit minus the units digit is one. If the integer is divisible by 11, what is the integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3542",
"style": "rule"
} | {
"index": "1241",
"split": "train"
} |
math | [
{
"content": "Determine into how many parts a regular tetrahedron is divided by six planes, each passing through one edge and the midpoint of the opposite edge of the tetrahedron. Find the volume of each part if the volume of the tetrahedron is 1. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "1/24",
"style": "rule"
} | {
"index": "1242",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $(1-\\frac{1}{a+1})÷\\frac{a^2-2a+1}{a^2-1}$, where $a=-2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "1243",
"split": "train"
} |
math | [
{
"content": "The function $f : \\mathbb{R} \\to \\mathbb{R}$ satisfies\n\\[f(x) + 2f(1 - x) = 3x^2\\]for all $x.$ Find $f(4).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "1244",
"split": "train"
} |
math | [
{
"content": "Side $AB$ of regular hexagon $ABCDEF$ is extended past $B$ to point $X$ such that $AX = 3AB$. Given that each side of the hexagon is $2$ units long, what is the length of segment $FX$? Express your answer in simplest radical form. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "2\\sqrt{13}",
"style": "rule"
} | {
"index": "1245",
"split": "train"
} |
math | [
{
"content": "A palindrome is an integer that reads the same forward and backward, such as 3663. What percent of the palindromes between 100 and 500 contain at least one 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10\\%",
"style": "rule"
} | {
"index": "1246",
"split": "train"
} |
math | [
{
"content": "A line with slope equal to $1$ and a line with slope equal to $2$ intersect at the point $P(1,6),$ as shown. [asy]\nunitsize(0.5 cm);\ndraw((0,-1)--(0,10),EndArrow);\ndraw((-10,0)--(5,0),EndArrow);\ndraw((-6,-1)--(5,10),linewidth(0.8));\ndraw((-2.5,-1)--(3,10),linewidth(0.8));\nlabel(\"$x$\",(5,0)... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "1247",
"split": "train"
} |
math | [
{
"content": "Let ${ a\\uparrow\\uparrow b = {{{{{a^{a}}^a}^{\\dots}}}^{a}}^{a}} $ , where there are $ b $ a's in total. That is $ a\\uparrow\\uparrow b $ is given by the recurrence \\[ a\\uparrow\\uparrow b = \\begin{cases} a & b=1 a^{a\\uparrow\\uparrow (b-1)} & b\\ge2\\end{cases} \\] What is the remai... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "1248",
"split": "train"
} |
math | [
{
"content": "Complex number $\\omega$ satisfies $\\omega^{5}=2$. Find the sum of all possible values of $\\omega^{4}+\\omega^{3}+\\omega^{2}+\\omega+1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "1249",
"split": "train"
} |
math | [
{
"content": "If the system of inequalities about $x$ is $\\left\\{{\\begin{array}{l}{-2({x-2})-x＜2}\\\\{\\frac{{k-x}}{2}≥-\\frac{1}{2}+x}\\end{array}}\\right.$ has at most $2$ integer solutions, and the solution to the one-variable linear equation about $y$ is $3\\left(y-1\\right)-2\\left(y-k\\right)=7$, deter... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "1250",
"split": "train"
} |
math | [
{
"content": "For how many integers \\(n\\) with \\(1 \\le n \\le 2020\\) is the product\n\\[\n \\prod_{k=0}^{n-1} \\left( \\left( 1 + e^{2 \\pi i k / n} \\right)^n + 1 \\right)\n\\]\nequal to zero? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "337",
"style": "rule"
} | {
"index": "1251",
"split": "train"
} |
math | [
{
"content": "Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible? Pleas... | {
"ground_truth": "132",
"style": "rule"
} | {
"index": "1252",
"split": "train"
} |
math | [
{
"content": "Given points $A(\\sin\\theta, 1)$, $B(\\cos\\theta, 0)$, $C(-\\sin\\theta, 2)$, and $\\overset{→}{AB}=\\overset{→}{BP}$.\n(I) Consider the function $f\\left(\\theta\\right)=\\overset{→}{BP}\\cdot\\overset{→}{CA}$, $\\theta\\in\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$, discuss the monotonic... | {
"ground_truth": "\\frac{\\sqrt{74}}{5}",
"style": "rule"
} | {
"index": "1253",
"split": "train"
} |
math | [
{
"content": "Find the roots of $6x^4+19x^3-51x^2+20x$. Enter your answer as a list of numbers separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0, \\frac{1}{2}, \\frac{4}{3}, -5",
"style": "rule"
} | {
"index": "1254",
"split": "train"
} |
math | [
{
"content": "To monitor the skier's movements, the coach divided the track into three sections of equal length. It was found that the skier's average speeds on these three separate sections were different. The time required for the skier to cover the first and second sections together was 40.5 minutes, and for... | {
"ground_truth": "58.5",
"style": "rule"
} | {
"index": "1255",
"split": "train"
} |
math | [
{
"content": "In the triangular pyramid $P-ABC$, $PA \\perp$ the base $ABC$, $AB=1$, $AC=2$, $\\angle BAC=60^{\\circ}$, the volume is $\\frac{\\sqrt{3}}{3}$, then the volume of the circumscribed sphere of the triangular pyramid is $\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.$ Please output the final answer within \\boxed{}... | {
"ground_truth": "\\frac{8 \\sqrt{2}}{3} \\pi",
"style": "rule"
} | {
"index": "1256",
"split": "train"
} |
math | [
{
"content": "In a factory, Erika assembles 3 calculators in the same amount of time that Nick assembles 2 calculators. Also, Nick assembles 1 calculator in the same amount of time that Sam assembles 3 calculators. How many calculators in total can be assembled by Nick, Erika, and Sam in the same amount of time... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "1257",
"split": "train"
} |
math | [
{
"content": "Find the four-digit number that is a perfect square, where the thousands digit is the same as the tens digit, and the hundreds digit is 1 greater than the units digit. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8281",
"style": "rule"
} | {
"index": "1258",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle{PQR}$ be a right triangle with $PQ = 90$, $PR = 120$, and $QR = 150$. Let $C_{1}$ be the inscribed circle. Construct $\\overline{ST}$ with $S$ on $\\overline{PR}$ and $T$ on $\\overline{QR}$, such that $\\overline{ST}$ is perpendicular to $\\overline{PR}$ and tangent to $C_{1}$. Co... | {
"ground_truth": "725",
"style": "rule"
} | {
"index": "1259",
"split": "train"
} |
math | [
{
"content": "Six consecutive integers are written on a blackboard. When one of them is erased, the sum of the remaining five integers is 2012. What is the sum of the digits of the integer that was erased? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "1260",
"split": "train"
} |
math | [
{
"content": "Consider a polynomial with integer coefficients given by:\n\\[8x^5 + b_4 x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 24 = 0.\\]\nFind the number of different possible rational roots of this polynomial. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "1261",
"split": "train"
} |
math | [
{
"content": "The cubic polynomial $q(x)$ satisfies $q(1) = 5,$ $q(6) = 20,$ $q(14) = 12,$ and $q(19) = 30.$ Find\n\\[q(0) + q(1) + q(2) + \\dots + q(20).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "357",
"style": "rule"
} | {
"index": "1262",
"split": "train"
} |
math | [
{
"content": "In an Olympic 100-meter final, there are 10 sprinters competing, among which 4 are Americans. The gold, silver, and bronze medals are awarded to first, second, and third place, respectively. Calculate the number of ways the medals can be awarded if at most two Americans are to receive medals. Plea... | {
"ground_truth": "588",
"style": "rule"
} | {
"index": "1263",
"split": "train"
} |
math | [
{
"content": "Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. She starts with a triangle $A_{0} A_{1} A_{2}$ where angle $A_{0}$ is $90^{\\circ}$, angle $A_{1}$ is $60^{\\circ}$, and $A_{0} A_{1}$ is 1. She then extends the pasture. First, she extends $A_{2} A_{0}$ to $A_{3}$ such t... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "1264",
"split": "train"
} |
math | [
{
"content": "Given $y_1 = x^2 - 7x + 6$, $y_2 = 7x - 3$, and $y = y_1 + xy_2$, find the value of $y$ when $x = 2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "1265",
"split": "train"
} |
math | [
{
"content": "A gardener wants to plant 3 maple trees, 4 oak trees, and 5 birch trees in a row. He will randomly determine the order of these trees. What is the probability that no two birch trees are adjacent? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7/99",
"style": "rule"
} | {
"index": "1266",
"split": "train"
} |
math | [
{
"content": "In the fall of 1996, a total of 800 students participated in an annual school clean-up day. The organizers of the event expect that in each of the years 1997, 1998, and 1999, participation will increase by 50% over the previous year. The number of participants the organizers will expect in the f... | {
"ground_truth": "2700",
"style": "rule"
} | {
"index": "1267",
"split": "train"
} |
math | [
{
"content": "Given that Gill leaves Lille at 09:00, the train travels the first 27 km at 96 km/h and then stops at Lens for 3 minutes before traveling the final 29 km to Lillers at 96 km/h, calculate the arrival time at Lillers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "09:38",
"style": "rule"
} | {
"index": "1268",
"split": "train"
} |
math | [
{
"content": "On an island, there are knights who always tell the truth and liars who always lie. At the main celebration, 100 islanders sat around a large round table. Half of the attendees said the phrase: \"both my neighbors are liars,\" while the remaining said: \"among my neighbors, there is exactly one li... | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "1269",
"split": "train"
} |
math | [
{
"content": "Anna and Aaron walk along paths formed by the edges of a region of squares. How far did they walk in total? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "640 \\text{ m}",
"style": "rule"
} | {
"index": "1270",
"split": "train"
} |
math | [
{
"content": "A \"clearance game\" has the following rules: in the $n$-th round, a die is rolled $n$ times. If the sum of the points from these $n$ rolls is greater than $2^n$, then the player clears the round. Questions:\n(1) What is the maximum number of rounds a player can clear in this game?\n(2) What is th... | {
"ground_truth": "\\frac{100}{243}",
"style": "rule"
} | {
"index": "1271",
"split": "train"
} |
math | [
{
"content": "Find the last 3 digits of \\(1 \\times 3 \\times 5 \\times 7 \\times \\cdots \\times 2005\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "375",
"style": "rule"
} | {
"index": "1272",
"split": "train"
} |
math | [
{
"content": "We are given $2n$ natural numbers\n\\[1, 1, 2, 2, 3, 3, \\ldots, n - 1, n - 1, n, n.\\]\nFind all $n$ for which these numbers can be arranged in a row such that for each $k \\leq n$, there are exactly $k$ numbers between the two numbers $k$. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "$n=3,4,7,8$",
"style": "rule"
} | {
"index": "1273",
"split": "train"
} |
math | [
{
"content": "What is the radius of the smallest circle inside which every planar closed polygonal line with a perimeter of 1 can be enclosed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "1274",
"split": "train"
} |
math | [
{
"content": "In the triangular prism $P-ABC$, the vertex $P$ projects orthogonally onto the circumcenter $O$ of the base $\\triangle ABC$, and the midpoint of $PO$ is $M$. Construct a cross-section $\\alpha$ through $AM$ parallel to $BC$, and denote $\\angle PAM$ as $\\theta_1$. The acute dihedral angle betwee... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "1275",
"split": "train"
} |
math | [
{
"content": "If $\\sqrt{x+2}=2$, then $(x+2)^2$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "1276",
"split": "train"
} |
math | [
{
"content": "A square is inscribed in an equilateral triangle such that each vertex of the square touches the perimeter of the triangle. One side of the square intersects and forms a smaller equilateral triangle within which we inscribe another square in the same manner, and this process continues infinitely. ... | {
"ground_truth": "\\frac{3 - \\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "1277",
"split": "train"
} |
math | [
{
"content": "Mark and Sandy are walking to the convenience store at the midpoint of their coordinates. Mark is standing at $(0,7)$, and Sandy is standing at $(-4,-1)$. At what coordinate will they meet? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-2,3)",
"style": "rule"
} | {
"index": "1278",
"split": "train"
} |
math | [
{
"content": "Given sets $A=\\{1,3,5,7,9\\}$ and $B=\\{0,3,6,9,12\\}$, then $A\\cap (\\complement_{\\mathbb{N}} B) = \\_\\_\\_\\_\\_\\_$; the number of proper subsets of $A\\cup B$ is $\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "255",
"style": "rule"
} | {
"index": "1279",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the two points $(0, -\\sqrt{3})$ and $(0, \\sqrt{3})$ equals $4$. Let the trajectory of point $P$ be $C$.\n\n$(1)$ Write the equation of $C$;\n\n$(2)$ Suppose the line $y=kx+1$ intersects $C$ at points $A$ and $B$. Fo... | {
"ground_truth": "\\frac{4\\sqrt{65}}{17}",
"style": "rule"
} | {
"index": "1280",
"split": "train"
} |
math | [
{
"content": "Menkara has a $4 \\times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? Please output the final answer... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "1281",
"split": "train"
} |
math | [
{
"content": "Given two vectors in the plane, $\\mathbf{a} = (2m+1, 3)$ and $\\mathbf{b} = (2, m)$, and $\\mathbf{a}$ is in the opposite direction to $\\mathbf{b}$, calculate the magnitude of $\\mathbf{a} + \\mathbf{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "1282",
"split": "train"
} |
math | [
{
"content": "A sports league has 20 teams split into four divisions of 5 teams each. How many games are there in a complete season if each team plays every other team in its own division three times and every team in the other divisions once? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "270",
"style": "rule"
} | {
"index": "1283",
"split": "train"
} |
math | [
{
"content": "The brothers found a treasure of gold and silver. They divided it so that each got 100 kg. The eldest got the most gold - 25 kg - and one-eighth of all the silver. How much gold was in the treasure? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "1284",
"split": "train"
} |
math | [
{
"content": "Find the constant term in the expansion of \\\\((x+ \\frac {2}{x}+1)^{6}\\\\) (Answer with a numerical value) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "581",
"style": "rule"
} | {
"index": "1285",
"split": "train"
} |
math | [
{
"content": "A national team needs to select 4 out of 6 sprinters to participate in the 4×100 m relay at the Asian Games. If sprinter A cannot run the first leg and sprinter B cannot run the fourth leg, there are a total of ______ ways to participate. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "252",
"style": "rule"
} | {
"index": "1286",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $\\Gamma$: $\\dfrac {x^{2}}{a^{2}}+y^{2}=1(a > 1)$, its left focus is $F_{1}$, the right vertex is $A_{1}$, and the top vertex is $B_{1}$. The circle $P$ that passes through points $F_{1}$, $A_{1}$, and $B_{1}$ has its center coordinates at $\\left( \\dfrac { \\sqrt {3}- \\sqrt {... | {
"ground_truth": "\\dfrac { \\sqrt {3}}{2}",
"style": "rule"
} | {
"index": "1287",
"split": "train"
} |
math | [
{
"content": "What is the smallest integer $x$ for which $x<2x-7$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "1288",
"split": "train"
} |
math | [
{
"content": "What is the value of $(9! \\cdot 5! \\cdot 2!) / (8! \\cdot 6!)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "1289",
"split": "train"
} |
math | [
{
"content": "If it costs two cents for each plastic digit used to number each locker and it costs $294.94 to label all lockers up to a certain number, calculate the highest locker number labeled. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3963",
"style": "rule"
} | {
"index": "1290",
"split": "train"
} |
math | [
{
"content": "If $f(x)$ is a function, then we define the function $f^{(n)}(x)$ to be the result of $n$ applications of $f$ to $x$, where $n$ is a positive integer. For example, $f^{(3)}(x)=f(f(f(x)))$.\n\nWe define the $\\textit{order}$ of an input $x$ with respect to $f$ to be the smallest positive integer $m... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "1291",
"split": "train"
} |
math | [
{
"content": "How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "1292",
"split": "train"
} |
math | [
{
"content": "If $n$ is a positive integer, then let $f(n)$ be the sum of the digits of $\\frac{1}{5^{{}^n}}$ that are to the right of the decimal point. What is the smallest positive integer $n$ such that $f(n) > 10$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "1293",
"split": "train"
} |
math | [
{
"content": "A \"pass level game\" has the following rules: On the \\( n \\)-th level, a die is thrown \\( n \\) times. If the sum of the points that appear in these \\( n \\) throws is greater than \\( 2^{n} \\), then the player passes the level. \n\n1. What is the maximum number of levels a person can pass i... | {
"ground_truth": "\\frac{100}{243}",
"style": "rule"
} | {
"index": "1294",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), point \\(O\\) is the center of the circumcircle, and point \\(L\\) is the midpoint of side \\(AB\\). The circumcircle of triangle \\(ALO\\) intersects the line \\(AC\\) at point \\(K\\). Find the area of triangle \\(ABC\\) if \\(\\angle LOA = 45^\\circ\\), \\(LK = 8\\), and ... | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "1295",
"split": "train"
} |
math | [
{
"content": "The center of a hyperbola is at the origin, the foci are on the coordinate axes, and the distance from point \\( P(-2,0) \\) to its asymptote is \\( \\frac{2 \\sqrt{6}}{3} \\). If a line with slope \\( \\frac{\\sqrt{2}}{2} \\) passes through point \\( P \\) and intersects the hyperbola at points \... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "1296",
"split": "train"
} |
math | [
{
"content": "What is the value of $\\log_{10}{16} + 3\\log_{5}{25} + 4\\log_{10}{2} + \\log_{10}{64} - \\log_{10}{8}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9.311",
"style": "rule"
} | {
"index": "1297",
"split": "train"
} |
math | [
{
"content": "The product of three positive integers $a$, $b$, and $c$ equals 1176. What is the minimum possible value of the sum $a + b + c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "59",
"style": "rule"
} | {
"index": "1298",
"split": "train"
} |
math | [
{
"content": "Definition: If the line $l$ is tangent to the graphs of the functions $y=f(x)$ and $y=g(x)$, then the line $l$ is called the common tangent line of the functions $y=f(x)$ and $y=g(x)$. If the functions $f(x)=a\\ln x (a>0)$ and $g(x)=x^{2}$ have exactly one common tangent line, then the value of th... | {
"ground_truth": "2e",
"style": "rule"
} | {
"index": "1299",
"split": "train"
} |
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