data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Find the number of functions $f(x)$ from $\\{1, 2, 3, 4, 5\\}$ to $\\{1, 2, 3, 4, 5\\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\\{1, 2, 3, 4, 5\\}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "756",
"style": "rule"
} | {
"index": "0",
"split": "train"
} |
math | [
{
"content": "Given that Jo climbs a flight of 8 stairs, and Jo can take the stairs 1, 2, 3, or 4 at a time, or a combination of these steps that does not exceed 4 with a friend, determine the number of ways Jo and the friend can climb the stairs together, including instances where either Jo or the friend climb... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "1",
"split": "train"
} |
math | [
{
"content": "The positive four-digit integers that use each of the four digits $1,$ $2,$ $3,$ and $4$ exactly once are ordered from least to greatest. What is the $15^{\\text{th}}$ integer in the list? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3214",
"style": "rule"
} | {
"index": "2",
"split": "train"
} |
math | [
{
"content": "Convert \\(531_8\\) to base 7. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1002_7",
"style": "rule"
} | {
"index": "3",
"split": "train"
} |
math | [
{
"content": "Find the largest natural number \\( n \\) with the following property: for any odd prime number \\( p \\) less than \\( n \\), the difference \\( n - p \\) is also a prime number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "4",
"split": "train"
} |
math | [
{
"content": "Find the sum of the infinite series $\\frac{1}{3^{2}-1^{2}}\\left(\\frac{1}{1^{2}}-\\frac{1}{3^{2}}\\right)+\\frac{1}{5^{2}-3^{2}}\\left(\\frac{1}{3^{2}}-\\frac{1}{5^{2}}\\right)+\\frac{1}{7^{2}-5^{2}}\\left(\\frac{1}{5^{2}}-\\frac{1}{7^{2}}\\right)+$ Please output the final answer within \\boxed{... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "5",
"split": "train"
} |
math | [
{
"content": "Xiao Zhang has three watches. The first watch runs 2 minutes fast every hour, the second watch runs 6 minutes fast, and the third watch runs 16 minutes fast. If the minute hands of the three watches are currently all pointing in the same direction, after how many hours will the three minute hands ... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "6",
"split": "train"
} |
math | [
{
"content": "Given that $1 \\leq x \\leq 2$, calculate the number of different integer values of the expression $10-10|2x-3|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "7",
"split": "train"
} |
math | [
{
"content": "If $b$ and $c$ are constants and $(x + 2)(x + b) = x^2 + cx + 6$, then $c$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$5$",
"style": "rule"
} | {
"index": "8",
"split": "train"
} |
math | [
{
"content": "Calculate $180 \\div \\left( 12 + 9 \\times 3 - 4 \\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{36}{7}",
"style": "rule"
} | {
"index": "9",
"split": "train"
} |
math | [
{
"content": "Consider a dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square, which is at a distance of 1 unit from the center, contains 8 unit squares. The second ring, at a distance of 2 units from the center, contains 16 unit squares. If the pa... | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "10",
"split": "train"
} |
math | [
{
"content": "$A$ and $B$ plays a game, with $A$ choosing a positive integer $n \\in \\{1, 2, \\dots, 1001\\} = S$ . $B$ must guess the value of $n$ by choosing several subsets of $S$ , then $A$ will tell $B$ how many subsets $n$ is in. $B$ will do this three times selecting $k_1, k_2$ then... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "11",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6, BC=5=DA,$and $CD=4.$ Draw circles of radius 3 centered at $A$ and $B,$ and circles of radius 2 centered at $C$ and $D.$ A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\\frac{-k+m\\s... | {
"ground_truth": "134",
"style": "rule"
} | {
"index": "12",
"split": "train"
} |
math | [
{
"content": "Determine the minimum possible value of the sum\n\\[\n\\frac{a}{3b} + \\frac{b}{6c} + \\frac{c}{9a},\n\\]\nwhere \\(a, b, c\\) are positive real numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 \\cdot \\frac{1}{\\sqrt[3]{162}}",
"style": "rule"
} | {
"index": "13",
"split": "train"
} |
math | [
{
"content": "For a real number $x$, define $\\heartsuit(x)$ to be the average of $x$ and $x^2$. What is $\\heartsuit(1)+\\heartsuit(2)+\\heartsuit(3)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "14",
"split": "train"
} |
math | [
{
"content": "An isosceles triangle, a square, and a regular pentagon each have a perimeter of 20 inches. What is the ratio of the side length of the triangle (assuming both equal sides for simplicity) to the side length of the square? Express your answer as a common fraction. Please output the final answer wit... | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "15",
"split": "train"
} |
math | [
{
"content": "Simplify\n\\[(1 + \\cot A - \\csc A)(1 + \\tan A + \\sec A).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "16",
"split": "train"
} |
math | [
{
"content": "Find the number of permutations $(a_1, a_2, a_3, a_4, a_5, a_6, a_7)$ of $(1,2,3,4,5,6,7)$ that satisfy\n\\[\\frac{a_1 + 1}{2} \\cdot \\frac{a_2 + 2}{2} \\cdot \\frac{a_3 + 3}{2} \\cdot \\frac{a_4 + 4}{2} \\cdot \\frac{a_5 + 5}{2} \\cdot \\frac{a_6 + 6}{2} \\cdot \\frac{a_7 + 7}{2} > 7!.\\] Please... | {
"ground_truth": "5039",
"style": "rule"
} | {
"index": "17",
"split": "train"
} |
math | [
{
"content": "The difference in the areas of two similar triangles is $18$ square feet, and the ratio of the larger area to the smaller is the square of an integer. The area of the smaller triangle, in square feet, is an integer, and one of its sides is $3$ feet. The corresponding side of the larger triangle,... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "18",
"split": "train"
} |
math | [
{
"content": "It is known that the numbers $\\frac{x}{2}, 2x - 3, \\frac{18}{x} + 1$, taken in the specified order, form a geometric progression. Find the common ratio of this progression. Round your answer to two decimal places. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.08",
"style": "rule"
} | {
"index": "19",
"split": "train"
} |
math | [
{
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{\\pi} 2^{4} \\cdot \\sin^{2}\\left(\\frac{x}{2}\\right) \\cos^{6}\\left(\\frac{x}{2}\\right) \\, dx\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5 \\pi}{8}",
"style": "rule"
} | {
"index": "20",
"split": "train"
} |
math | [
{
"content": "Each edge of a regular tetrahedron is given a stripe. The choice of which edge to stripe is made at random. What is the probability that there is at least one triangle face with all its edges striped? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1695}{4096}",
"style": "rule"
} | {
"index": "21",
"split": "train"
} |
math | [
{
"content": "A cube is suspended in space with its top and bottom faces horizontal. The cube has one top face, one bottom face, and four side faces. Determine the number of ways to move from the top face to the bottom face, visiting each face at most once, without moving directly from the top face to the botto... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "22",
"split": "train"
} |
math | [
{
"content": "What is the area of the region enclosed by $x^2 + y^2 = |x| - |y|$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "23",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer $n$ such that $2n$ is a perfect square and $3n$ is a perfect cube? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "24",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer solution to $\\tan{19x^{\\circ}}=\\dfrac{\\cos{96^{\\circ}}+\\sin{96^{\\circ}}}{\\cos{96^{\\circ}}-\\sin{96^{\\circ}}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "159",
"style": "rule"
} | {
"index": "25",
"split": "train"
} |
math | [
{
"content": "What is the sum of the digits of the integer which is equal to \\(6666666^{2} - 3333333^{2}\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "26",
"split": "train"
} |
math | [
{
"content": "Given the line $l: 4x+3y-8=0$ passes through the center of the circle $C: x^2+y^2-ax=0$ and intersects circle $C$ at points A and B, with O as the origin.\n(I) Find the equation of circle $C$.\n(II) Find the equation of the tangent to circle $C$ at point $P(1, \\sqrt {3})$.\n(III) Find the area of... | {
"ground_truth": "\\frac{16}{5}",
"style": "rule"
} | {
"index": "27",
"split": "train"
} |
math | [
{
"content": "Let \\\\(f(x)=ax^{2}-b\\sin x\\\\) and \\\\(f′(0)=1\\\\), \\\\(f′\\left( \\dfrac {π}{3}\\right)= \\dfrac {1}{2}\\\\). Find the values of \\\\(a\\\\) and \\\\(b\\\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "28",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{m}=(\\sin A, \\frac {1}{2})$ and $\\overrightarrow{n}=(3,\\sin A+ \\sqrt {3}\\cos A)$ are collinear, where $A$ is an internal angle of $\\triangle ABC$. \n$(1)$ Find the size of angle $A$; \n$(2)$ If $BC=2$, find the maximum value of the area $S$ of $\\triangle AB... | {
"ground_truth": "\\sqrt {3}",
"style": "rule"
} | {
"index": "29",
"split": "train"
} |
math | [
{
"content": "A \\(10 \\times 1\\) rectangular pavement is to be covered by tiles which are either green or yellow, each of width 1 and of varying integer lengths from 1 to 10. Suppose you have an unlimited supply of tiles for each color and for each of the varying lengths. How many distinct tilings of the rect... | {
"ground_truth": "1022",
"style": "rule"
} | {
"index": "30",
"split": "train"
} |
math | [
{
"content": "Given an isolated ground state iron atom with atomic number Z = 26, determine the number of orbitals that contain one or more electrons. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "31",
"split": "train"
} |
math | [
{
"content": "Sides $\\overline{AH}$ and $\\overline{CD}$ of regular octagon $ABCDEFGH$ are extended to meet at point $P$. What is the degree measure of angle $P$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45^\\circ",
"style": "rule"
} | {
"index": "32",
"split": "train"
} |
math | [
{
"content": "What is the value of $\\frac{14!}{5!9!}$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2002",
"style": "rule"
} | {
"index": "33",
"split": "train"
} |
math | [
{
"content": "We say that a number of 20 digits is *special* if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10^9 - 1",
"style": "rule"
} | {
"index": "34",
"split": "train"
} |
math | [
{
"content": "For $x$ real, the inequality $1 \\le |x-2| \\le 7$ is equivalent to Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$-5 \\le x \\le 1$ or $3 \\le x \\le 9$",
"style": "rule"
} | {
"index": "35",
"split": "train"
} |
math | [
{
"content": "Let $n$ be the answer to this problem. The polynomial $x^{n}+ax^{2}+bx+c$ has real coefficients and exactly $k$ real roots. Find the sum of the possible values of $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "36",
"split": "train"
} |
math | [
{
"content": "$A, B, C, D,$ and $E$ are collinear in that order such that $AB = BC = 1, CD = 2,$ and $DE = 9$. If $P$ can be any point in space, what is the smallest possible value of $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "37",
"split": "train"
} |
math | [
{
"content": "Let \\( S = \\{1, 2, \\ldots, 2005\\} \\). If any set of \\( n \\) pairwise co-prime numbers in \\( S \\) always contains at least one prime number, what is the minimum value of \\( n \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "38",
"split": "train"
} |
math | [
{
"content": "Given that $0 < α < \\dfrac {π}{2}$, $- \\dfrac {π}{2} < β < 0$, $\\cos ( \\dfrac {π}{4}+α)= \\dfrac {1}{3}$, and $\\cos ( \\dfrac {π}{4}-β)= \\dfrac { \\sqrt {3}}{3}$, find $\\cos (α+β)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac {5 \\sqrt {3}}{9}",
"style": "rule"
} | {
"index": "39",
"split": "train"
} |
math | [
{
"content": "Eight students from a university are preparing to carpool for a trip. There are two students from each grade level—freshmen, sophomores, juniors, and seniors—divided into two cars, Car A and Car B, with each car seating four students. The seating arrangement of the four students in the same car is... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "40",
"split": "train"
} |
math | [
{
"content": "What is the value of \\(1.90 \\frac{1}{1-\\sqrt[4]{3}}+\\frac{1}{1+\\sqrt[4]{3}}+\\frac{2}{1+\\sqrt{3}}\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "41",
"split": "train"
} |
math | [
{
"content": "Two circles have radii 13 and 30, and their centers are 41 units apart. The line through the centers of the two circles intersects the smaller circle at two points; let $A$ be the one outside the larger circle. Suppose $B$ is a point on the smaller circle and $C$ a point on the larger circle such ... | {
"ground_truth": "12 \\sqrt{13}",
"style": "rule"
} | {
"index": "42",
"split": "train"
} |
math | [
{
"content": "Lucy has $19$ dollars and $23$ cents. She wants to buy as many popsicles as she can with her money. The popsicles are priced at $1$ dollar and $60$ cents each. How many popsicles can she buy? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "43",
"split": "train"
} |
math | [
{
"content": "Find the length of a line segment from a vertex to the center of a regular hexagon with a side length of 12. Express your answer in simplest radical form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "44",
"split": "train"
} |
math | [
{
"content": "A square in the $xy$-plane has area $A$, and three of its vertices have $x$-coordinates 2, 0, and 18 in some order. Find the sum of all possible values of $A$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1168",
"style": "rule"
} | {
"index": "45",
"split": "train"
} |
math | [
{
"content": "The line $2x+ay-2=0$ is parallel to the line $ax+(a+4)y-4=0$. Find the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "46",
"split": "train"
} |
math | [
{
"content": "Point P lies on the curve represented by the equation $$\\sqrt {(x-5)^{2}+y^{2}}- \\sqrt {(x+5)^{2}+y^{2}}=6$$. If the y-coordinate of point P is 4, then its x-coordinate is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "x = -3\\sqrt{2}",
"style": "rule"
} | {
"index": "47",
"split": "train"
} |
math | [
{
"content": "Let $(x, y)$ be a pair of real numbers satisfying $$56x+33y=\\frac{-y}{x^{2}+y^{2}}, \\quad \\text { and } \\quad 33x-56y=\\frac{x}{x^{2}+y^{2}}$$ Determine the value of $|x|+|y|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{65}",
"style": "rule"
} | {
"index": "48",
"split": "train"
} |
math | [
{
"content": "For the Olympic torch relay, it is planned to select 6 cities from 8 in a certain province to establish the relay route, satisfying the following conditions. How many methods are there for each condition?\n(1) Only one of the two cities, A and B, is selected. How many methods are there? How many d... | {
"ground_truth": "19440",
"style": "rule"
} | {
"index": "49",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\lfloor-5.77\\rfloor+\\lceil-3.26\\rceil+\\lfloor15.93\\rfloor+\\lceil32.10\\rceil$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "50",
"split": "train"
} |
math | [
{
"content": "A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes t... | {
"ground_truth": "840",
"style": "rule"
} | {
"index": "51",
"split": "train"
} |
math | [
{
"content": "Calculate:<br/>$(1)(\\sqrt{\\frac{1}{3}})^{2}+\\sqrt{0．{3}^{2}}-\\sqrt{\\frac{1}{9}}$;<br/>$(2)(\\sqrt{6}-\\sqrt{\\frac{1}{2}})-(\\sqrt{24}+2\\sqrt{\\frac{2}{3}})$;<br/>$(3)(\\frac{\\sqrt{32}}{3}-4\\sqrt{\\frac{1}{2}}+3\\sqrt{27})÷2\\sqrt{2}$;<br/>$(4)(\\sqrt{3}+\\sqrt{2}-1)(\\sqrt{3}-\\sqrt{2}+1)... | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "52",
"split": "train"
} |
math | [
{
"content": "In triangle $PQR$, $PQ = 8$, $QR = 15$, $PR = 17$, and $QS$ is the angle bisector. Find the length of $QS$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{87.04}",
"style": "rule"
} | {
"index": "53",
"split": "train"
} |
math | [
{
"content": "In the base 10 arithmetic problem $H M M T+G U T S=R O U N D$, each distinct letter represents a different digit, and leading zeroes are not allowed. What is the maximum possible value of $R O U N D$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16352",
"style": "rule"
} | {
"index": "54",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{P}$ be the matrix for projecting onto the vector $\\begin{pmatrix} 4 \\\\ -7 \\end{pmatrix}.$ Find $\\det \\mathbf{P}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "55",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\ln (ax+1)+ \\frac {x^{3}}{3}-x^{2}-ax(a∈R)$,\n(1) Find the range of values for the real number $a$ such that $y=f(x)$ is an increasing function on $[4,+∞)$;\n(2) When $a\\geqslant \\frac {3 \\sqrt {2}}{2}$, let $g(x)=\\ln [x^{2}(ax+1)]+ \\frac {x^{3}}{3}-3ax-f(x)(x > 0)$... | {
"ground_truth": "\\ln 2- \\frac {2}{3}",
"style": "rule"
} | {
"index": "56",
"split": "train"
} |
math | [
{
"content": "The front tires of a car wear out after 25,000 km, and the rear tires wear out after 15,000 km. When is it advisable to swap the tires so that they wear out equally? (Assume that the tires are swapped only once, although in practice drivers do this more frequently.) Please output the final answer ... | {
"ground_truth": "9375",
"style": "rule"
} | {
"index": "57",
"split": "train"
} |
math | [
{
"content": "For all positive integers $n$ , let $f(n)$ return the smallest positive integer $k$ for which $\\tfrac{n}{k}$ is not an integer. For example, $f(6) = 4$ because $1$ , $2$ , and $3$ all divide $6$ but $4$ does not. Determine the largest possible value of $f(n)$ as $n$ ranges ... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "58",
"split": "train"
} |
math | [
{
"content": "Pentagon ABCDE has a vertical line of symmetry. What is the $y$-coordinate of vertex C so that the area of the pentagon is 40 square units? [asy]\nunitsize(2mm);\ndefaultpen(linewidth(.7pt)+fontsize(8pt));\ndotfactor=4;\n\npair A=(0,0), B=(0,4), C=(2,16), D=(4,4), E=(4,0);\npair[] dots={A,B,C,D,E}... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "59",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be a triangle with $A B=5, B C=8, C A=11$. The incircle $\\omega$ and $A$-excircle $^{1} \\Gamma$ are centered at $I_{1}$ and $I_{2}$, respectively, and are tangent to $B C$ at $D_{1}$ and $D_{2}$, respectively. Find the ratio of the area of $\\triangle A I_{1} D_{1}$ to the area of $\... | {
"ground_truth": "\\frac{1}{9}",
"style": "rule"
} | {
"index": "60",
"split": "train"
} |
math | [
{
"content": "A line \\( l \\), parallel to the diagonal \\( AC_1 \\) of a unit cube \\( ABCDA_1B_1C_1D_1 \\), is equidistant from the lines \\( BD \\), \\( A_1D_1 \\), and \\( CB_1 \\). Find the distances from the line \\( l \\) to these lines. Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "\\frac{\\sqrt{2}}{6}",
"style": "rule"
} | {
"index": "61",
"split": "train"
} |
math | [
{
"content": "If $(x+2)(x-3)=14$, find the product of the possible values of $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-20",
"style": "rule"
} | {
"index": "62",
"split": "train"
} |
math | [
{
"content": "In quadrilateral $ABCD$, $\\angle A = 120^\\circ$, and $\\angle B$ and $\\angle D$ are right angles. Given $AB = 13$ and $AD = 46$, find the length of $AC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "62",
"style": "rule"
} | {
"index": "63",
"split": "train"
} |
math | [
{
"content": "Line $l_{1}$: $mx+2y-3=0$ is parallel to line $l_{2}$: $3x+\\left(m-1\\right)y+m-6=0$. Find the value of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "64",
"split": "train"
} |
math | [
{
"content": "In each part of this problem, cups are arranged in a circle and numbered \\(1, 2, 3, \\ldots\\). A ball is placed in cup 1. Then, moving clockwise around the circle, a ball is placed in every \\(n\\)th cup. The process ends when cup 1 contains two balls. \n\n(a) There are 12 cups in the circle and... | {
"ground_truth": "360",
"style": "rule"
} | {
"index": "65",
"split": "train"
} |
math | [
{
"content": "For the curve $ C: y = \\frac {1}{1 + x^2}$ , Let $ A(\\alpha ,\\ f(\\alpha)),\\ B\\left( - \\frac {1}{\\alpha},\\ f\\left( - \\frac {1}{\\alpha} \\right)\\right)\\ (\\alpha > 0).$ Find the minimum area bounded by the line segments $ OA,\\ OB$ and $ C,$ where $ O$ is the origin.\n\nNote t... | {
"ground_truth": "\\frac{\\pi}{2} - \\frac{1}{2}",
"style": "rule"
} | {
"index": "66",
"split": "train"
} |
math | [
{
"content": "In spherical coordinates, the point $\\left( 3, \\frac{2 \\pi}{7}, \\frac{8 \\pi}{5} \\right)$ is equivalent to what other point, in the standard spherical coordinate representation? Enter your answer in the form $(\\rho,\\theta,\\phi),$ where $\\rho > 0,$ $0 \\le \\theta < 2 \\pi,$ and $0 \\le \... | {
"ground_truth": "\\left( 3, \\frac{9 \\pi}{7}, \\frac{2 \\pi}{5} \\right)",
"style": "rule"
} | {
"index": "67",
"split": "train"
} |
math | [
{
"content": "Find the remainder when \\( 8x^4 - 18x^3 - 6x^2 + 4x - 30 \\) is divided by \\( 2x - 8 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "786",
"style": "rule"
} | {
"index": "68",
"split": "train"
} |
math | [
{
"content": "Compute the sum of the squares of the roots of the equation \\[x\\sqrt{x} - 8x + 9\\sqrt{x} - 2 = 0,\\] given that all of the roots are real and nonnegative. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "69",
"split": "train"
} |
math | [
{
"content": "Given a finite sequence $S=(a_1,a_2,\\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence \n$\\left(\\frac{a_1+a_2}{2},\\frac{a_2+a_3}{2},\\ldots ,\\frac{a_{n-1}+a_n}{2}\\right)$\nof $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\\le m\\le n-1$, define $A^m(S)=A(A^... | {
"ground_truth": "\\sqrt{2}-1",
"style": "rule"
} | {
"index": "70",
"split": "train"
} |
math | [
{
"content": "Estimate the year in which the population of Nisos will be approximately 6,000. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2075",
"style": "rule"
} | {
"index": "71",
"split": "train"
} |
math | [
{
"content": "Six 6-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{16}",
"style": "rule"
} | {
"index": "72",
"split": "train"
} |
math | [
{
"content": "$A, B, C, D, E, F, G$ are seven people sitting around a circular table. If $d$ is the total number of ways that $B$ and $G$ must sit next to $C$, find the value of $d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "73",
"split": "train"
} |
math | [
{
"content": "There is a board of size 7×12 cells and a cube, the side of which is equal to a cell. One face of the cube is painted with non-drying paint. The cube can be placed in a certain cell of the board and rolled over an edge to an adjacent face. The cube cannot be placed twice in the same cell. What is ... | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "74",
"split": "train"
} |
math | [
{
"content": "A builder has two identical bricks. She places them side by side in three different ways, resulting in shapes with surface areas of 72, 96, and 102. What is the surface area of one original brick?\nA) 36\nB) 48\nC) 52\nD) 54\nE) 60 Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "75",
"split": "train"
} |
math | [
{
"content": "In the complex plane, the points \\( 0, z, \\frac{1}{z}, z+\\frac{1}{z} \\) form a parallelogram with an area of \\( \\frac{35}{37} \\). If the real part of \\( z \\) is greater than 0, find the minimum value of \\( \\left| z + \\frac{1}{z} \\right| \\). Please output the final answer within \\box... | {
"ground_truth": "\\frac{5 \\sqrt{74}}{37}",
"style": "rule"
} | {
"index": "76",
"split": "train"
} |
math | [
{
"content": "When studying the operation of a new type of cyclic thermal engine, it was found that during part of the period it receives heat, and the absolute power of heat supply is expressed by the law: \n\\[ P_{1}(t)=P_{0} \\frac{\\sin (\\omega t)}{100+\\sin (t^{2})}, \\quad 0<t<\\frac{\\pi}{\\omega}. \\] ... | {
"ground_truth": "1/3",
"style": "rule"
} | {
"index": "77",
"split": "train"
} |
math | [
{
"content": "One caterer charges a basic fee of $\\$100$ plus $\\$15$ per person. A second caterer charges a basic fee of $\\$200$ plus $\\$12$ per person. What is the least number of people for which the second caterer is cheaper? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "78",
"split": "train"
} |
math | [
{
"content": "Given that $O$ is the coordinate origin, the complex numbers $z_1$ and $z_2$ correspond to the vectors $\\overrightarrow{OZ_1}$ and $\\overrightarrow{OZ_2}$, respectively. $\\bar{z_1}$ is the complex conjugate of $z_1$. The vectors are represented as $\\overrightarrow{OZ_1} = (10 - a^2, \\frac{1}{... | {
"ground_truth": "\\frac{\\sqrt{130}}{8}",
"style": "rule"
} | {
"index": "79",
"split": "train"
} |
math | [
{
"content": "Shuxin begins with 10 red candies, 7 yellow candies, and 3 blue candies. After eating some of the candies, there are equal numbers of red, yellow, and blue candies remaining. What is the smallest possible number of candies that Shuxin ate? Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "80",
"split": "train"
} |
math | [
{
"content": "Given that the sum of two numbers is 15 and their product is 36, find the sum of their reciprocals and the sum of their squares. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "153",
"style": "rule"
} | {
"index": "81",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer that leaves a remainder of 4 when divided by 5 and a remainder of 6 when divided by 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "82",
"split": "train"
} |
math | [
{
"content": "If $a$, $b$, $c$, and $d$ are four positive numbers whose product is 1, calculate the minimum value of the algebraic expression $a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "83",
"split": "train"
} |
math | [
{
"content": "The area of an equilateral triangle is numerically equal to the length of one of its sides. What is the perimeter of the triangle, in units? Express your answer in simplest radical form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "84",
"split": "train"
} |
math | [
{
"content": "Alexia designed a logo 2 inches wide and 1.5 inches tall to be used on her school's notebooks. The school wants the logo on the notebook to be 8 inches wide. How tall, in inches, will it be if it is enlarged proportionately? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "85",
"split": "train"
} |
math | [
{
"content": "An ellipse is defined parametrically by\n\\[(x,y) = \\left( \\frac{2 (\\sin t - 1)}{2 - \\cos t}, \\frac{3 (\\cos t - 5)}{2 - \\cos t} \\right).\\]Then the equation of the ellipse can be written in the form\n\\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\\]where $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ are integ... | {
"ground_truth": "1381",
"style": "rule"
} | {
"index": "86",
"split": "train"
} |
math | [
{
"content": "A square is initially divided into sixteen equal smaller squares. The center four squares are then each divided into sixteen smaller squares of equal area, and this pattern continues indefinitely. In each set of sixteen smaller squares, the four corner squares are shaded. What fractional part of t... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "87",
"split": "train"
} |
math | [
{
"content": "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that $f(1) = 1$ and\n\\[f(xy + f(x)) = xf(y) + f(x)\\]for all real numbers $x$ and $y.$\n\nLet $n$ be the number of possible values of $f \\left( \\frac{1}{2} \\right),$ and let $s$ be the sum of all possible values of $f \\left( \\frac{1}{... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "88",
"split": "train"
} |
math | [
{
"content": "Let $F_n$ be the $n$th Fibonacci number, where as usual $F_1 = F_2 = 1$ and $F_{n + 1} = F_n + F_{n - 1}.$ Then\n\\[\\prod_{k = 2}^{100} \\left( \\frac{F_k}{F_{k - 1}} - \\frac{F_k}{F_{k + 1}} \\right) = \\frac{F_a}{F_b}\\]for some positive integers $a$ and $b.$ Enter the ordered pair $(a,b).$ P... | {
"ground_truth": "(100,101)",
"style": "rule"
} | {
"index": "89",
"split": "train"
} |
math | [
{
"content": "If $5$ lunks can be traded for $3$ kunks, and $2$ kunks will buy $4$ apples, how many lunks are needed to purchase one dozen apples? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "90",
"split": "train"
} |
math | [
{
"content": "Sarah buys $3$ gumballs from a gumball machine that contains $10$ orange, $6$ green, and $9$ yellow gumballs. What is the probability that the first gumball is orange, the second is green or yellow, and the third is also orange? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "9/92",
"style": "rule"
} | {
"index": "91",
"split": "train"
} |
math | [
{
"content": "Given that the random variable $ξ$ follows a normal distribution $N(1,σ^{2})$, and $P(ξ\\leqslant 4)=0.79$, determine the value of $P(-2\\leqslant ξ\\leqslant 1)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.29",
"style": "rule"
} | {
"index": "92",
"split": "train"
} |
math | [
{
"content": "A right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "93",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\\frac{\\sqrt{5}}{5}$, and its right latus rectum equation is $x=5$.\n(1) Find the equation of the ellipse;\n(2) A line $l$ with a slope of $1$ passes through the right focus $F$ of the ellips... | {
"ground_truth": "\\frac{16\\sqrt{10}}{9}",
"style": "rule"
} | {
"index": "94",
"split": "train"
} |
math | [
{
"content": "There are eleven positive integers $n$ such that there exists a convex polygon with $n$ sides whose angles, in degrees, are unequal integers that are in arithmetic progression. Find the sum of these values of $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "106",
"style": "rule"
} | {
"index": "95",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = 3x^2 - 7$ and $g(f(4)) = 9$. What is $g(f(-4))$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "96",
"split": "train"
} |
math | [
{
"content": "Evaluate \\begin{align*}\n\\left(c^c-c(c-1)^c\\right)^c\n\\end{align*} when $c=3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "97",
"split": "train"
} |
math | [
{
"content": "The number of positive integers $k$ for which the equation\n\\[kx-12=3k\\]has an integer solution for $x$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "98",
"split": "train"
} |
math | [
{
"content": "To stabilize housing prices, a local government decided to build a batch of affordable housing for the community. The plan is to purchase a piece of land for 16 million yuan and build a residential community with 10 buildings on it. Each building has the same number of floors, and each floor has a... | {
"ground_truth": "1225",
"style": "rule"
} | {
"index": "99",
"split": "train"
} |
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