data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relative prime. Find the maximum value of $n $ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "6600",
"split": "train"
} |
math | [
{
"content": "Points \\(P, Q, R,\\) and \\(S\\) lie in the plane of the square \\(EFGH\\) such that \\(EPF\\), \\(FQG\\), \\(GRH\\), and \\(HSE\\) are equilateral triangles. If \\(EFGH\\) has an area of 25, find the area of quadrilateral \\(PQRS\\). Express your answer in simplest radical form. Please output th... | {
"ground_truth": "100 + 50\\sqrt{3}",
"style": "rule"
} | {
"index": "6601",
"split": "train"
} |
math | [
{
"content": "On a table, there are five clocks with hands. It is allowed to move any number of them forward. For each clock, the time by which it is moved will be referred to as the translation time. It is required to set all clocks such that they show the same time. What is the minimum total translation time ... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "6602",
"split": "train"
} |
math | [
{
"content": "What is the value of $x^5 - 5x$ when $x = 4$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1004",
"style": "rule"
} | {
"index": "6603",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a_n\\}$ where $a_1=1$ and $a_n=70$ (for $n\\geq3$), find all possible values of $n$ if the common difference is a natural number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "6604",
"split": "train"
} |
math | [
{
"content": "To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. For her small kite, Genevieve draws the kite on a grid with the points one inch apart, as shown below.\n\n[asy]\nfor ( int x = 0; x <= 6; ++x )\n{\n\nfor ( int y = 0; y <= 7; +... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "6605",
"split": "train"
} |
math | [
{
"content": "Let $(b_1,b_2,b_3,\\ldots,b_{14})$ be a permutation of $(1,2,3,\\ldots,14)$ for which\n$b_1>b_2>b_3>b_4>b_5>b_6>b_7>b_8 \\mathrm{\\ and \\ } b_8<b_9<b_{10}<b_{11}<b_{12}<b_{13}<b_{14}.$\nFind the number of such permutations. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "1716",
"style": "rule"
} | {
"index": "6606",
"split": "train"
} |
math | [
{
"content": "The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-16",
"style": "rule"
} | {
"index": "6607",
"split": "train"
} |
math | [
{
"content": "When the number of repeated experiments is large enough, probability can be estimated using frequency. The mathematician Pearson once tossed a fair coin 24,000 times in an experiment. The number of times the coin landed heads up was 12,012 times, with a frequency of 0.5005. Therefore, the probabil... | {
"ground_truth": "0.5005",
"style": "rule"
} | {
"index": "6608",
"split": "train"
} |
math | [
{
"content": "Simplify the expression: $$\\dfrac{\\sqrt{450}}{\\sqrt{288}} + \\dfrac{\\sqrt{245}}{\\sqrt{96}}.$$ Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{30 + 7\\sqrt{30}}{24}",
"style": "rule"
} | {
"index": "6609",
"split": "train"
} |
math | [
{
"content": "Given the four equations:\n$\\textbf{(1)}\\ 3y-2x=12 \\qquad\\textbf{(2)}\\ -2x-3y=10 \\qquad\\textbf{(3)}\\ 3y+2x=12 \\qquad\\textbf{(4)}\\ 2y+3x=10$\nThe pair representing the perpendicular lines is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\text{(1) and (4)}",
"style": "rule"
} | {
"index": "6610",
"split": "train"
} |
math | [
{
"content": "Given a sequence \\( a_1, a_2, a_3, \\ldots, a_n \\) of non-zero integers such that the sum of any 7 consecutive terms is positive and the sum of any 11 consecutive terms is negative, what is the largest possible value for \\( n \\)? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "6611",
"split": "train"
} |
math | [
{
"content": "The difference between two positive integers is 8 and their product is 56. What is the sum of these integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12\\sqrt{2}",
"style": "rule"
} | {
"index": "6612",
"split": "train"
} |
math | [
{
"content": "Don has a bottle of milk with $\\frac{1}{5}$ of a gallon of milk in it. Don's friend Rachel drinks $\\frac{2}{3}$ of the milk he has. How many gallons of milk did Rachel drink? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{15}",
"style": "rule"
} | {
"index": "6613",
"split": "train"
} |
math | [
{
"content": "In a certain city, license plate numbers consist of 6 digits (ranging from 0 to 9), but it is required that any two plates differ by at least 2 digits (for example, license plates 038471 and 030471 cannot both be used). Determine the maximum number of different license plates that can be issued an... | {
"ground_truth": "100000",
"style": "rule"
} | {
"index": "6614",
"split": "train"
} |
math | [
{
"content": "Let $ S $ be the set of all sides and diagonals of a regular pentagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\tfrac{4}{9}",
"style": "rule"
} | {
"index": "6615",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $(xOy)$, the parametric equations of line $l$ are given by $\\begin{cases}x=1+\\frac{\\sqrt{2}}{2}t\\\\y=\\frac{\\sqrt{2}}{2}t\\end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with the origin $O$ as the pole and the $x$-axis as the po... | {
"ground_truth": "\\sqrt{2}-1",
"style": "rule"
} | {
"index": "6616",
"split": "train"
} |
math | [
{
"content": "There is a polynomial $P(x)$ with integer coefficients such that\\[P(x)=\\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}\\]holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "220",
"style": "rule"
} | {
"index": "6617",
"split": "train"
} |
math | [
{
"content": "Brachycephalus frogs have three toes on each foot and two fingers on each hand. The common frog has five toes on each foot and four fingers on each hand. Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toes and 92 fingers. Ho... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "6618",
"split": "train"
} |
math | [
{
"content": "In tetrahedron $ABCD$, $AD > AB$, $AD \\perp AB$, $AD \\perp AC$, $\\angle BAC = \\frac{\\pi}{3}$. Let the areas of triangles $ADB$, $ADC$, $ABC$, and $BCD$ be $S_{1}$, $S_{2}$, $S_{3}$, and $S_{4}$, respectively. It is known that $S_{1} + S_{2} = S_{3} + S_{4}$. Find the value of $\\frac{S_{3}}{S... | {
"ground_truth": "3/2",
"style": "rule"
} | {
"index": "6619",
"split": "train"
} |
math | [
{
"content": "Every day, from Monday to Friday, an old man went to the blue sea and cast his net into the sea. Each day, he caught no more fish than the previous day. In total, he caught exactly 100 fish over the five days. What is the minimum total number of fish he could have caught on Monday, Wednesday, and ... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "6620",
"split": "train"
} |
math | [
{
"content": "Solve for $x$: $\\frac{5x}{(x+3)} - \\frac3{(x+3)} = \\frac1{(x+3)}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "6621",
"split": "train"
} |
math | [
{
"content": "Given that $α$ is an internal angle of a triangle, and $\\sin α + \\cos α = \\frac{1}{5}$. \n$(1)$ Find the value of $\\tan α$; \n$(2)$ Find the value of $\\frac{\\sin(\\frac{3π}{2}+α)\\sin(\\frac{π}{2}-α)\\tan^{3}(π-α)}{\\cos(\\frac{π}{2}+α)\\cos(\\frac{3π}{2}-α)}$. Please output the final answ... | {
"ground_truth": "-\\frac{4}{3}",
"style": "rule"
} | {
"index": "6622",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{r}$ and $\\mathbf{s}$ be two three-dimensional unit vectors such that the angle between them is $45^\\circ$. Find the area of the parallelogram whose diagonals correspond to $\\mathbf{r} + 3\\mathbf{s}$ and $3\\mathbf{r} + \\mathbf{s}$. Please output the final answer within \\boxed{}... | {
"ground_truth": "\\frac{3\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "6623",
"split": "train"
} |
math | [
{
"content": "The numbers \\(1, 2, 3, \\ldots, 7\\) are randomly divided into two non-empty subsets. What is the probability that the sum of the numbers in the two subsets is equal? If the probability is expressed as \\(\\frac{p}{q}\\) in its lowest terms, find \\(p + q\\). Please output the final answer within... | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "6624",
"split": "train"
} |
math | [
{
"content": "The sum of two numbers is $12$ and their difference is $20$. What is the smaller number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "6625",
"split": "train"
} |
math | [
{
"content": "If $x = \\sqrt{1+\\sqrt{1+\\sqrt{1+\\sqrt{1+\\cdots}}}}$, then: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 < x < 2",
"style": "rule"
} | {
"index": "6626",
"split": "train"
} |
math | [
{
"content": "Suppose there are 3 counterfeit coins of equal weight mixed with 12 genuine coins. All counterfeit coins weigh differently from the genuine coins. A pair of coins is selected at random without replacement from the 15 coins, followed by selecting a second pair from the remaining 13 coins. The combi... | {
"ground_truth": "\\frac{15}{19}",
"style": "rule"
} | {
"index": "6627",
"split": "train"
} |
math | [
{
"content": "Find all real $ a$, such that there exist a function $ f: \\mathbb{R}\\rightarrow\\mathbb{R}$ satisfying the following inequality:\n\\[ x\\plus{}af(y)\\leq y\\plus{}f(f(x))\n\\]\nfor all $ x,y\\in\\mathbb{R}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "a < 0 \\text{ or } a = 1",
"style": "rule"
} | {
"index": "6628",
"split": "train"
} |
math | [
{
"content": "In a trapezoid, the lengths of the diagonals are known to be 6 and 8, and the length of the midline is 5. Find the height of the trapezoid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.8",
"style": "rule"
} | {
"index": "6629",
"split": "train"
} |
math | [
{
"content": "How many integers between 0 and 8 inclusive have an inverse modulo 9? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "6630",
"split": "train"
} |
math | [
{
"content": "There are 4 different colors of light bulbs, with each color representing a different signal. Assuming there is an ample supply of each color, we need to install one light bulb at each vertex $P, A, B, C, A_{1}, B_{1}, C_{1}$ of an airport signal tower (as shown in the diagram), with the condition... | {
"ground_truth": "2916",
"style": "rule"
} | {
"index": "6631",
"split": "train"
} |
math | [
{
"content": "A man and his faithful dog simultaneously started moving along the perimeter of a block from point \\( A \\) at time \\( t_{0} = 0 \\) minutes. The man moved at a constant speed clockwise, while the dog ran at a constant speed counterclockwise. It is known that the first time they met was \\( t_{1... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "6632",
"split": "train"
} |
math | [
{
"content": "Let $g(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $g(2)=0$, $90<g(9)<100$, $120<g(10)<130$, $7000k<g(150)<7000(k+1)$ for some integer $k$. What is $k$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "k=6",
"style": "rule"
} | {
"index": "6633",
"split": "train"
} |
math | [
{
"content": "In right triangle $DEF$, where $DE = 15$, $DF = 9$, and $EF = 12$ units, find the distance from $F$ to the midpoint of segment $DE$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7.5",
"style": "rule"
} | {
"index": "6634",
"split": "train"
} |
math | [
{
"content": "The greatest common divisor of 21 and some number between 50 and 60 is 7. What is the number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "6635",
"split": "train"
} |
math | [
{
"content": "If one vertex of an ellipse and its two foci form the vertices of an equilateral triangle, then find the eccentricity $e$ of the ellipse. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{2}",
"style": "rule"
} | {
"index": "6636",
"split": "train"
} |
math | [
{
"content": "Find the number ot 6-tuples $(x_1, x_2,...,x_6)$ , where $x_i=0,1 or 2$ and $x_1+x_2+...+x_6$ is even Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "365",
"style": "rule"
} | {
"index": "6637",
"split": "train"
} |
math | [
{
"content": "In how many different ways can André form exactly \\( \\$10 \\) using \\( \\$1 \\) coins, \\( \\$2 \\) coins, and \\( \\$5 \\) bills? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "6638",
"split": "train"
} |
math | [
{
"content": "Five times $A$'s money added to $B$'s money is more than $51.00$. Three times $A$'s money minus $B$'s money is $21.00$.\nIf $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, then: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$a>9, b>6$",
"style": "rule"
} | {
"index": "6639",
"split": "train"
} |
math | [
{
"content": "If the ratio of the surface areas of three spheres is 1:4:9, then the ratio of their volumes is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 : 8 : 27",
"style": "rule"
} | {
"index": "6640",
"split": "train"
} |
math | [
{
"content": "What is the maximum number of queens that can be placed on the black squares of an $8 \\times 8$ chessboard so that each queen is attacked by at least one of the others? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "6641",
"split": "train"
} |
math | [
{
"content": "Select 5 people from 4 boys and 5 girls to participate in a math extracurricular group. How many different ways are there to select under the following conditions?\n\n (1) Select 2 boys and 3 girls, and girl A must be selected;\n\n (2) Select at most 4 girls, and boy A and girl B cannot be selecte... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "6642",
"split": "train"
} |
math | [
{
"content": "Points \\( A, B, C \\) are situated sequentially, with the distance \\( AB \\) being 3 km and the distance \\( BC \\) being 4 km. A cyclist departed from point \\( A \\) heading towards point \\( C \\). Simultaneously, a pedestrian departed from point \\( B \\) heading towards point \\( A \\). It ... | {
"ground_truth": "2.1",
"style": "rule"
} | {
"index": "6643",
"split": "train"
} |
math | [
{
"content": "What is the greatest common factor of 68 and 92? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "6644",
"split": "train"
} |
math | [
{
"content": "In the magical forest of Santa Claus, cedars grow one and a half times taller than firs and grow for 9 hours. Firs grow for 2 hours. Santa Claus planted cedar seeds at 12 o'clock and fir seeds at 2 o'clock in the afternoon. What time was it when the trees were of the same height? (The trees grow u... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "6645",
"split": "train"
} |
math | [
{
"content": "Given non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2|\\overrightarrow{b}|$, and $(\\overrightarrow{a}-\\overrightarrow{b})\\bot \\overrightarrow{b}$, calculate the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Please output the fi... | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "6646",
"split": "train"
} |
math | [
{
"content": "A natural number, when raised to the sixth power, has digits which, when arranged in ascending order, are:\n\n$$\n0,2,3,4,4,7,8,8,9\n$$\n\nWhat is this number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "6647",
"split": "train"
} |
math | [
{
"content": "Find\n\\[\\sum_{N = 1}^{1024} \\lfloor \\log_2 N \\rfloor.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8204",
"style": "rule"
} | {
"index": "6648",
"split": "train"
} |
math | [
{
"content": "The second and fifth terms of an arithmetic sequence are 17 and 19, respectively. What is the eighth term? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "6649",
"split": "train"
} |
math | [
{
"content": "Given a tetrahedron \\(A B C D\\) with the following edge lengths: \\(A B = 2\\), \\(C D = 2\\sqrt{5}\\), \\(A C = B D = 3\\), and \\(A D = B C = \\sqrt{5}\\), find the volume of the tetrahedron. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4/3",
"style": "rule"
} | {
"index": "6650",
"split": "train"
} |
math | [
{
"content": "If $x$ and $y$ are positive integers such that $xy - 5x + 6y = 119$, what is the minimal possible value of $|x - y|$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "77",
"style": "rule"
} | {
"index": "6651",
"split": "train"
} |
math | [
{
"content": "The acute angles of a right triangle are $a^{\\circ}$ and $b^{\\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "6652",
"split": "train"
} |
math | [
{
"content": "Let \\( O \\) be an interior point of \\( \\triangle ABC \\). Extend \\( AO \\) to meet \\( BC \\) at \\( D \\). Similarly, extend \\( BO \\) and \\( CO \\) to meet \\( CA \\) and \\( AB \\) respectively at \\( E \\) and \\( F \\). Given \\( AO = 30 \\), \\( FO = 20 \\), \\( BO = 60 \\), \\( DO = ... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "6653",
"split": "train"
} |
math | [
{
"content": "An equilateral triangle has two vertices at $(0,5)$ and $(8,5)$. If the third vertex is in the first quadrant, what is the y-coordinate? Express your answer in simplest radical form. [asy]\ndraw((-1,0)--(11,0),Arrows);\ndraw((0,-1)--(0,12),Arrows);\nfor(int i=0;i<11;++i)\n{draw((i,-0.1)--(i,0.1));... | {
"ground_truth": "5+4\\sqrt{3}",
"style": "rule"
} | {
"index": "6654",
"split": "train"
} |
math | [
{
"content": "What is the smallest $n$ for which there exists an $n$-gon that can be divided into a triangle, quadrilateral, ..., up to a 2006-gon? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2006",
"style": "rule"
} | {
"index": "6655",
"split": "train"
} |
math | [
{
"content": "Given a finite sequence $S=(2, 2x, 2x^2,\\ldots ,2x^{200})$ of $n=201$ real numbers, let $A(S)$ be the sequence $\\left(\\frac{a_1+a_2}{2},\\frac{a_2+a_3}{2},\\ldots ,\\frac{a_{200}+a_{201}}{2}\\right)$ of $n-1=200$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\\le m\\le 150$, d... | {
"ground_truth": "2^{3/8} - 1",
"style": "rule"
} | {
"index": "6656",
"split": "train"
} |
math | [
{
"content": "The sale ad read: \"Buy three tires at the regular price and get the fourth tire for 3 dollars.\" Sam paid 240 dollars for a set of four tires at the sale. What was the regular price of one tire? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "79",
"style": "rule"
} | {
"index": "6657",
"split": "train"
} |
math | [
{
"content": "A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\\frac{1}{3}$, independently of its p... | {
"ground_truth": "252",
"style": "rule"
} | {
"index": "6658",
"split": "train"
} |
math | [
{
"content": "What is the maximum area that a rectangle can have if the coordinates of its vertices satisfy the equation \\( |y-x| = (y+x+1)(5-x-y) \\), and its sides are parallel to the lines \\( y = x \\) and \\( y = -x \\)? Give the square of the value of the maximum area found as the answer. (12 points) Ple... | {
"ground_truth": "432",
"style": "rule"
} | {
"index": "6659",
"split": "train"
} |
math | [
{
"content": "Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{7}{64}",
"style": "rule"
} | {
"index": "6660",
"split": "train"
} |
math | [
{
"content": "Let\n\\[p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.\\]\n Suppose that\n\\[p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1)\\\\ = p(1,1) = p(1, - 1) = p(2,2) = 0.\\]\n There is a point $\\left(\\frac {a}{c},\\frac {b}{c}\\right)$ for which $p\\left(\\... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "6661",
"split": "train"
} |
math | [
{
"content": "Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion. \n\nRule 1: If the integer is less than 10, multiply it by 9.\nRule 2: If the integer is even and greater t... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "6662",
"split": "train"
} |
math | [
{
"content": "The area of square \\(ABCD\\) is 64. The midpoints of its sides are joined to form the square \\(EFGH\\). The midpoints of its sides are \\(J, K, L,\\) and \\(M\\). The area of the shaded region is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "6663",
"split": "train"
} |
math | [
{
"content": "In the given triangle $ABC$, construct the points $C_{1}$ on side $AB$ and $A_{1}$ on side $BC$ such that the intersection point $P$ of lines $AA_{1}$ and $CC_{1}$ satisfies $AP / PA_{1} = 3 / 2$ and $CP / PC_{1} = 2 / 1$. In what ratio does point $C_{1}$ divide side $AB$? Please output the final ... | {
"ground_truth": "2/3",
"style": "rule"
} | {
"index": "6664",
"split": "train"
} |
math | [
{
"content": "Point $A$ lies on the line $y = \\frac{12}{5} x - 3$, and point $B$ lies on the parabola $y = x^2$. What is the minimum length of the segment $AB$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.6",
"style": "rule"
} | {
"index": "6665",
"split": "train"
} |
math | [
{
"content": "Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos? Please output the final answer with... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "6666",
"split": "train"
} |
math | [
{
"content": "a) Find all the divisors of the number 572 based on its prime factorization.\n\nb) How many divisors does the number \\(572 a^{3} b c\\) have if:\n\n I. \\(a, b\\), and \\(c\\) are prime numbers greater than 20 and different from each other? \n\n II. \\(a = 31\\), \\(b = 32\\), and \\(c = 33\\... | {
"ground_truth": "384",
"style": "rule"
} | {
"index": "6667",
"split": "train"
} |
math | [
{
"content": "Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\\dfrac{1}{2}$ is $\\dfrac{a-b\\pi}{c}$, where $a$, $b$, and $c$ are positive integers with $\\gcd(a,b,c)=1$. W... | {
"ground_truth": "59",
"style": "rule"
} | {
"index": "6668",
"split": "train"
} |
math | [
{
"content": "Successive discounts of $10\\%$ and $20\\%$ are equivalent to a single discount of: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28\\%",
"style": "rule"
} | {
"index": "6669",
"split": "train"
} |
math | [
{
"content": "Let $p = 2017,$ a prime number. Let $N$ be the number of ordered triples $(a,b,c)$ of integers such that $1 \\le a,b \\le p(p-1)$ and $a^b-b^a=p \\cdot c$ . Find the remainder when $N$ is divided by $1000000.$ *Proposed by Evan Chen and Ashwin Sah*\n\n*Remark:* The problem was initiall... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "6670",
"split": "train"
} |
math | [
{
"content": "The diagonals of a rhombus measure 18 feet and 12 feet. What is the perimeter of the rhombus? Express your answer in simplest radical form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12\\sqrt{13}\\text{ feet}",
"style": "rule"
} | {
"index": "6671",
"split": "train"
} |
math | [
{
"content": "We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there?\... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "6672",
"split": "train"
} |
math | [
{
"content": "How many obtuse interior angles are in an obtuse triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "6673",
"split": "train"
} |
math | [
{
"content": "Given that there are 21 students in Dr. Smith's physics class, the average score before including Simon's project score was 86. After including Simon's project score, the average for the class rose to 88. Calculate Simon's score on the project. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "6674",
"split": "train"
} |
math | [
{
"content": "Solve the equation \\[-2x^2 = \\frac{4x + 2}{x + 2}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "6675",
"split": "train"
} |
math | [
{
"content": "Given a parallelepiped \\(A B C D A_{1} B_{1} C_{1} D_{1}\\), a point \\(X\\) is chosen on edge \\(A_{1} D_{1}\\), and a point \\(Y\\) is chosen on edge \\(B C\\). It is known that \\(A_{1} X = 5\\), \\(B Y = 3\\), and \\(B_{1} C_{1} = 14\\). The plane \\(C_{1} X Y\\) intersects the ray \\(D A\\) ... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "6676",
"split": "train"
} |
math | [
{
"content": "Given any two positive real numbers $x$ and $y$, then $x \\diamond y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x \\diamond y$ satisfies the equations \\((x \\cdot y) \\diamond y=x(y \\diamond y)\\) and \\((x \\diamond 1) \\diamond x=x \\d... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "6677",
"split": "train"
} |
math | [
{
"content": "In triangle \\( \\triangle ABC \\), \\( \\angle BAC = 90^\\circ \\), \\( AC = AB = 4 \\), and point \\( D \\) is inside \\( \\triangle ABC \\) such that \\( AD = \\sqrt{2} \\). Find the minimum value of \\( BD + CD \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{10}",
"style": "rule"
} | {
"index": "6678",
"split": "train"
} |
math | [
{
"content": "The function $y=(m^2-m-1)x^{m^2-3m-3}$ is a power function, and it is an increasing function on the interval $(0, +\\infty)$. Find the value of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "6679",
"split": "train"
} |
math | [
{
"content": "Twenty points are equally spaced around the circumference of a circle. Kevin draws all the possible chords that connect pairs of these points. How many of these chords are longer than the radius of the circle but shorter than its diameter? Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "6680",
"split": "train"
} |
math | [
{
"content": "Compute the decimal expansion of \\sqrt{\\pi}$. Your score will be \\min (23, k)$, where $k$ is the number of consecutive correct digits immediately following the decimal point in your answer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.77245385090551602729816 \\ldots",
"style": "rule"
} | {
"index": "6681",
"split": "train"
} |
math | [
{
"content": "Estimate the number of primes among the first thousand primes divide some term of the sequence\n \\[2^0+1,2^1+1,2^2+1,2^3+1,\\ldots.\\]\n An estimate of $E$ earns $2^{1-0.02|A-E|}$ points, where $A$ is the actual answer.\n\n\n*2021 CCA Math Bonanza Lightning Round #5.4* Please output t... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "6682",
"split": "train"
} |
math | [
{
"content": "Given $x, y, z \\in \\mathbb{R}$ and $x^{2}+y^{2}+z^{2}=25$, find the maximum and minimum values of $x-2y+2z$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-15",
"style": "rule"
} | {
"index": "6683",
"split": "train"
} |
math | [
{
"content": "Six rhombi of side length 1 are arranged as shown. What is the perimeter of this figure? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "6684",
"split": "train"
} |
math | [
{
"content": "What three-digit positive integer is one more than a multiple of 3, 4, 5, 6, and 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "421",
"style": "rule"
} | {
"index": "6685",
"split": "train"
} |
math | [
{
"content": "The base of isosceles $\\triangle ABC$ is $24$ and its area is $60$. What is the length of one of the congruent sides? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "6686",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\left\\{a_{n}\\right\\}$ defined by $a_{0} = \\frac{1}{2}$ and $a_{n+1} = a_{n} + \\frac{a_{n}^{2}}{2023}$ for $n=0,1,2,\\ldots$, find the integer $k$ such that $a_{k} < 1 < a_{k+1}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2023",
"style": "rule"
} | {
"index": "6687",
"split": "train"
} |
math | [
{
"content": "Let \\[f(x) =\n\\begin{cases}\n2x + 9 &\\text{if }x<-2, \\\\\n5-2x&\\text{if }x\\ge -2.\n\\end{cases}\n\\]Find $f(3).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "6688",
"split": "train"
} |
math | [
{
"content": "For how many positive integers \\( n \\) less than or equal to 500 is \n$$(\\cos t - i\\sin t)^n = \\cos nt - i\\sin nt$$ \ntrue for all real \\( t \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "6689",
"split": "train"
} |
math | [
{
"content": "22. Let the function $f : \\mathbb{Z} \\to \\mathbb{Z}$ take only integer inputs and have integer outputs. For any integers $x$ and $y$ , f satisfies $$ f(x) + f(y) = f(x + 1) + f(y - 1) $$ If $f(2016) = 6102$ and $f(6102) = 2016$ , what is $f(1)$ ?\n\n23. Let $d$ be a randomly chosen... | {
"ground_truth": "8117",
"style": "rule"
} | {
"index": "6690",
"split": "train"
} |
math | [
{
"content": "There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either p... | {
"ground_truth": "310",
"style": "rule"
} | {
"index": "6691",
"split": "train"
} |
math | [
{
"content": "If $f(x)=\\frac{x(x-1)}{2}$, then $f(x+2)$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{(x+2)f(x+1)}{x}",
"style": "rule"
} | {
"index": "6692",
"split": "train"
} |
math | [
{
"content": "Let $\\frac{x^2+y^2}{x^2-y^2} + \\frac{x^2-y^2}{x^2+y^2} = k$. Compute the following expression in terms of $k$: \n\\[ E(x,y) = \\frac{x^8 + y^8}{x^8-y^8} - \\frac{ x^8-y^8}{x^8+y^8}. \\]\n[i]Ciprus[/i] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{k^4 - 8k^2 + 16}{4k(k^2 + 4)}",
"style": "rule"
} | {
"index": "6693",
"split": "train"
} |
math | [
{
"content": "Given that the function f(x) = 2sinωx(cosωx + $\\sqrt{3}$sinωx) - $\\sqrt{3}$ (ω > 0) has a minimum positive period of π.\n(1) Find the interval where the function f(x) is monotonically increasing.\n(2) If the graph of function f(x) is shifted to the left by $\\frac{\\pi}{6}$ units and up by 2 uni... | {
"ground_truth": "\\frac{55\\pi}{4}",
"style": "rule"
} | {
"index": "6694",
"split": "train"
} |
math | [
{
"content": "In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\\circ$. The enclosed area of the hexagon is $6\\sqrt{3}$. What is the perimeter of the hexagon? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12\\sqrt{3}",
"style": "rule"
} | {
"index": "6695",
"split": "train"
} |
math | [
{
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence with common difference $d$, and $S_n$ is the sum of the first $n$ terms, if only $S_4$ is the minimum term among $\\{S_n\\}$, then the correct conclusion(s) can be drawn is/are ______.\n\n$(1) d > 0$ $(2) a_4 < 0$ $(3) a_5 > 0$ $(4) S_7 < 0$ ... | {
"ground_truth": "(1)(2)(3)(4)",
"style": "rule"
} | {
"index": "6696",
"split": "train"
} |
math | [
{
"content": "When a polynomial is divided by $2x^2 - 7x + 18,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0,1",
"style": "rule"
} | {
"index": "6697",
"split": "train"
} |
math | [
{
"content": "A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$, and that $f(x) = 1-|x-2|$ for $1\\le x \\le 3$. Find the smallest $x$ for which $f(x) = f(2001)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "429",
"style": "rule"
} | {
"index": "6698",
"split": "train"
} |
math | [
{
"content": "Alan, Jason, and Shervin are playing a game with MafsCounts questions. They each start with $2$ tokens. In each round, they are given the same MafsCounts question. The first person to solve the MafsCounts question wins the round and steals one token from each of the other players in the game. Th... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "6699",
"split": "train"
} |
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