data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In a collection of red, blue, and green marbles, there are $25\\%$ more red marbles than blue marbles, and there are $60\\%$ more green marbles than red marbles. Suppose that there are $r$ red marbles. What is the total number of marbles in the collection? Please output the final answer within \\b... | {
"ground_truth": "3.4r",
"style": "rule"
} | {
"index": "38900",
"split": "train"
} |
math | [
{
"content": "When the two-digit integer \\( XX \\), with equal digits, is multiplied by the one-digit integer \\( X \\), the result is the three-digit integer \\( PXQ \\). What is the greatest possible value of \\( PXQ \\) if \\( PXQ \\) must start with \\( P \\) and end with \\( X \\)? Please output the final... | {
"ground_truth": "396",
"style": "rule"
} | {
"index": "38901",
"split": "train"
} |
math | [
{
"content": "Given that the right focus of the ellipse $\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ is $F(\\sqrt{6},0)$, a line $l$ passing through $F$ intersects the ellipse at points $A$ and $B$. If the midpoint of chord $AB$ has coordinates $(\\frac{\\sqrt{6}}{3},-1)$, calculate the area of ... | {
"ground_truth": "12\\sqrt{3}\\pi",
"style": "rule"
} | {
"index": "38902",
"split": "train"
} |
math | [
{
"content": "Jenn randomly chooses a number $J$ from $1, 2, 3,\\ldots, 19, 20$. Bela then randomly chooses a number $B$ from $1, 2, 3,\\ldots, 19, 20$ distinct from $J$. The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\\frac{m}{n}$ where $m$ and $n$ are relatively pri... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "38903",
"split": "train"
} |
math | [
{
"content": "In the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$, \nand sides $BC$ and $ED$. Each side has length $1$. Also, $\\angle FAB = \\angle BCD = 60^\\circ$. \nThe area of the figure is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "38904",
"split": "train"
} |
math | [
{
"content": "Calculate:<br/>$(1)\\left(-12\\right)-5+\\left(-14\\right)-\\left(-39\\right)$;<br/>$(2)-2^{2}\\times 5-\\left(-12\\right)\\div 4-4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-21",
"style": "rule"
} | {
"index": "38905",
"split": "train"
} |
math | [
{
"content": "Find all triples of positive integers $(x, y, z)$ such that $x^{2}+y-z=100$ and $x+y^{2}-z=124$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(12,13,57)",
"style": "rule"
} | {
"index": "38906",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\frac{\\cos 2x}{\\sin(x+\\frac{π}{4})}$.\n(I) Find the domain of the function $f(x)$;\n(II) If $f(x)=\\frac{4}{3}$, find the value of $\\sin 2x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{9}",
"style": "rule"
} | {
"index": "38907",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, the parametric equations of the line $l$ are $\\left\\{{\\begin{array}{l}{x=4-\\frac{{\\sqrt{2}}}{2}t}\\\\{y=4+\\frac{{\\sqrt{2}}}{2}t}\\end{array}}\\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the origin $O$ as the pole an... | {
"ground_truth": "4 + 4\\sqrt{3}",
"style": "rule"
} | {
"index": "38908",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be a triangle with $A B=20, B C=10, C A=15$. Let $I$ be the incenter of $A B C$, and let $B I$ meet $A C$ at $E$ and $C I$ meet $A B$ at $F$. Suppose that the circumcircles of $B I F$ and $C I E$ meet at a point $D$ different from $I$. Find the length of the tangent from $A$ to the cir... | {
"ground_truth": "2 \\sqrt{30}",
"style": "rule"
} | {
"index": "38909",
"split": "train"
} |
math | [
{
"content": "Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $1001$, then what is the sum of the numbers written on the faces? Please output the fina... | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "38910",
"split": "train"
} |
math | [
{
"content": "Compute the sum of the number $10 - \\sqrt{2018}$ and its radical conjugate. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38911",
"split": "train"
} |
math | [
{
"content": "Given \\(0 \\leq x_0 < 1\\), let \n\\[\nx_n = \\left\\{ \\begin{array}{ll}\n2x_{n-1} & \\text{if } 2x_{n-1} < 1 \\\\\n2x_{n-1} - 1 & \\text{if } 2x_{n-1} \\geq 1\n\\end{array} \\right.\n\\]\nfor all integers \\(n > 0\\). Determine the number of initial values of \\(x_0\\) that satisfy \\(x_0 = x_6... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "38912",
"split": "train"
} |
math | [
{
"content": "The sum of seven consecutive even numbers is 686. What is the smallest of these seven numbers? Additionally, calculate the median and mean of this sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "98",
"style": "rule"
} | {
"index": "38913",
"split": "train"
} |
math | [
{
"content": "What is the value of $\frac{(20-16) \times (12+8)}{4}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38914",
"split": "train"
} |
math | [
{
"content": "A company has a total of 60 employees. In order to carry out club activities, a questionnaire survey was conducted among all employees. There are 28 people who like sports, 26 people who like literary and artistic activities, and 12 people who do not like either sports or literary and artistic act... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "38915",
"split": "train"
} |
math | [
{
"content": "Randomly and without replacement, select three numbers \\(a_1, a_2, a_3\\) from the set \\(\\{1, 2, \\cdots, 2014\\}\\). Then, from the remaining 2011 numbers, again randomly and without replacement, select three numbers \\(b_1, b_2, b_3\\). What is the probability that a brick with dimensions \\(... | {
"ground_truth": "1/4",
"style": "rule"
} | {
"index": "38916",
"split": "train"
} |
math | [
{
"content": "Players A and B participate in a two-project competition, with each project adopting a best-of-five format (the first player to win 3 games wins the match, and the competition ends), and there are no ties in each game. Based on the statistics of their previous matches, player A has a probability o... | {
"ground_truth": "\\frac{209}{162}",
"style": "rule"
} | {
"index": "38917",
"split": "train"
} |
math | [
{
"content": "A laboratory has $10$ experimental mice, among which $3$ have been infected with a certain virus, and the remaining $7$ are healthy. Random medical examinations are conducted one by one until all $3$ infected mice are identified. The number of different scenarios where the last infected mouse is d... | {
"ground_truth": "1512",
"style": "rule"
} | {
"index": "38918",
"split": "train"
} |
math | [
{
"content": "Let $G$ be the centroid of triangle $ABC.$ If $GA^2 + GB^2 + GC^2 = 58,$ then find $AB^2 + AC^2 + BC^2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "174",
"style": "rule"
} | {
"index": "38919",
"split": "train"
} |
math | [
{
"content": "In the isosceles triangle \\(ABC\\) with the sides \\(AB = BC\\), the angle \\(\\angle ABC\\) is \\(80^\\circ\\). Inside the triangle, a point \\(O\\) is taken such that \\(\\angle OAC = 10^\\circ\\) and \\(\\angle OCA = 30^\\circ\\). Find the angle \\(\\angle AOB\\). Please output the final answe... | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "38920",
"split": "train"
} |
math | [
{
"content": "Matěj had written six different natural numbers in a row in his notebook. The second number was double the first, the third was double the second, and similarly, each subsequent number was double the previous one. Matěj copied all these numbers into the following table in random order, one number ... | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "38921",
"split": "train"
} |
math | [
{
"content": "Given two dice are thrown in succession, with the six faces of each die labeled with $1$, $2$, $3$, $4$, $5$, and $6$ points, let event $A$ be \"$x+y$ is odd\" and event $B$ be \"$x$, $y$ satisfy $x+y<6$, where $x$ and $y$ are the numbers of points facing up on the two dice when they land on a tab... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "38922",
"split": "train"
} |
math | [
{
"content": "At what point does the line $3y-4x=12$ intersect the $x$-axis? Express your answer as an ordered pair. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-3,0)",
"style": "rule"
} | {
"index": "38923",
"split": "train"
} |
math | [
{
"content": "Let \\( n = \\overline{abc} \\) be a three-digit number, where \\( a, b, \\) and \\( c \\) are the digits of the number. If \\( a, b, \\) and \\( c \\) can form an isosceles triangle (including equilateral triangles), how many such three-digit numbers \\( n \\) are there? Please output the final a... | {
"ground_truth": "165",
"style": "rule"
} | {
"index": "38924",
"split": "train"
} |
math | [
{
"content": "Given an ellipse $E: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 \\left(a > b > 0\\right)$ that passes through the point $(0,1)$, and its eccentricity is $\\frac{\\sqrt{3}}{2}$.\n\n$(1)$ Find the standard equation of the ellipse $E$;\n\n$(2)$ Suppose a line $l: y = \\frac{1}{2}x + m$ intersects the el... | {
"ground_truth": "\\frac{\\sqrt{10}}{2}",
"style": "rule"
} | {
"index": "38925",
"split": "train"
} |
math | [
{
"content": "For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \\[20 + 10r + 10r^2 + 10r^3 + \\cdots.\\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=5040$. Find $T(b)+T(-b)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "504",
"style": "rule"
} | {
"index": "38926",
"split": "train"
} |
math | [
{
"content": "A regular octahedron is formed by joining the midpoints of the edges of a regular tetrahedron. Calculate the ratio of the volume of this octahedron to the volume of the original tetrahedron. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "38927",
"split": "train"
} |
math | [
{
"content": "$\\textbf{Problem 5.}$ Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $15/10$ minutes per day. \nIf you put them at the same correct time, What is the least number of days that must pass for both to give the correct time simultaneously? Please output the f... | {
"ground_truth": "1440",
"style": "rule"
} | {
"index": "38928",
"split": "train"
} |
math | [
{
"content": "A cake has a shape of triangle with sides $19,20$ and $21$ . It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't stick out of the plate. What is the minimal diameter of the plate? Please output the final answe... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "38929",
"split": "train"
} |
math | [
{
"content": "Jessica has exactly one of each of the first 30 states' new U.S. quarters. The quarters were released in the same order that the states joined the union. The graph below shows the number of states that joined the union in each decade. What fraction of Jessica's 30 coins represents states that join... | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "38930",
"split": "train"
} |
math | [
{
"content": "Two quadratic equations with unequal leading coefficients,\n$$\n(a-1) x^{2} - \\left(a^{2}+2\\right) x + \\left(a^{2}+2a\\right) = 0\n$$\nand\n$$\n(b-1) x^{2} - \\left(b^{2}+2\\right) x + \\left(b^{2}+2b\\right) = 0\n$$\n(where \\(a\\) and \\(b\\) are positive integers), have a common root. Find t... | {
"ground_truth": "256",
"style": "rule"
} | {
"index": "38931",
"split": "train"
} |
math | [
{
"content": "Two equal parallel chords are drawn $8$ inches apart in a circle of radius $8$ inches. The area of that part of the circle that lies between the chords is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$32\\sqrt{3}+21\\frac{1}{3}\\pi$",
"style": "rule"
} | {
"index": "38932",
"split": "train"
} |
math | [
{
"content": "Xiao Ming collected 20 pieces of data in a survey, as follows: \n$95\\ \\ \\ 91\\ \\ \\ 93\\ \\ \\ 95\\ \\ \\ 97\\ \\ \\ 99\\ \\ \\ 95\\ \\ \\ 98\\ \\ \\ 90\\ \\ \\ 99$ \n$96\\ \\ \\ 94\\ \\ \\ 95\\ \\ \\ 97\\ \\ \\ 96\\ \\ \\ 92\\ \\ \\ 94\\ \\ \\ 95\\ \\ \\ 96\\ \\ \\ 98$ \n$(1)$ When constru... | {
"ground_truth": "0.4",
"style": "rule"
} | {
"index": "38933",
"split": "train"
} |
math | [
{
"content": "A $6 \\times 9$ rectangle can be rolled to form two different cylinders. Calculate the ratio of the larger volume to the smaller volume. Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "38934",
"split": "train"
} |
math | [
{
"content": "Suppose \\(a\\), \\(b\\), and \\(c\\) are real numbers such that:\n\\[\n\\frac{ac}{a + b} + \\frac{ba}{b + c} + \\frac{cb}{c + a} = -12\n\\]\nand\n\\[\n\\frac{bc}{a + b} + \\frac{ca}{b + c} + \\frac{ab}{c + a} = 15.\n\\]\nCompute the value of:\n\\[\n\\frac{a}{a + b} + \\frac{b}{b + c} + \\frac{c}{... | {
"ground_truth": "-12",
"style": "rule"
} | {
"index": "38935",
"split": "train"
} |
math | [
{
"content": "Arrange 1, 2, 3, a, b, c in a row such that letter 'a' is not at either end and among the three numbers, exactly two are adjacent. The probability is $\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "38936",
"split": "train"
} |
math | [
{
"content": "Let $D$ be a regular ten-sided polygon with edges of length 1. A triangle $T$ is defined by choosing three vertices of $D$ and connecting them with edges. How many different (non-congruent) triangles $T$ can be formed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "38937",
"split": "train"
} |
math | [
{
"content": "There are 20 students in a class. In total, 10 of them have black hair, 5 of them wear glasses, and 3 of them both have black hair and wear glasses. How many of the students have black hair but do not wear glasses? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "38938",
"split": "train"
} |
math | [
{
"content": "Given $\\cos\\alpha =\\frac{\\sqrt{5}}{5}$ and $\\sin\\beta =\\frac{3\\sqrt{10}}{10}$, with $0 < \\alpha$, $\\beta < \\frac{\\pi}{2}$, determine the value of $\\alpha +\\beta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": "38939",
"split": "train"
} |
math | [
{
"content": "The number 770 is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either 40 or 41 from the number on the board. She performs moves until the number is not positive, and then she stops. Let $N$ be the number of sequences of moves that Melody could per... | {
"ground_truth": "318",
"style": "rule"
} | {
"index": "38940",
"split": "train"
} |
math | [
{
"content": "Let $(1+2x)^2(1-x)^5 = a + a_1x + a_2x^2 + \\ldots + a_7x^7$, then $a_1 - a_2 + a_3 - a_4 + a_5 - a_6 + a_7 =$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-31",
"style": "rule"
} | {
"index": "38941",
"split": "train"
} |
math | [
{
"content": "The set of vectors $\\left\\{ \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix}, \\begin{pmatrix} 3 \\\\ k \\end{pmatrix} \\right\\}$ is linearly dependent. Find all possible values of $k.$ Enter all the possible values, separated by commas. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "38942",
"split": "train"
} |
math | [
{
"content": "Evaluate the value of $3^2 \\times 4 \\times 6^3 \\times 7!$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39191040",
"style": "rule"
} | {
"index": "38943",
"split": "train"
} |
math | [
{
"content": "A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ part... | {
"ground_truth": "557",
"style": "rule"
} | {
"index": "38944",
"split": "train"
} |
math | [
{
"content": "Simplify the following expression: \\[2x+3x^2+1-(6-2x-3x^2).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6x^2+4x-5",
"style": "rule"
} | {
"index": "38945",
"split": "train"
} |
math | [
{
"content": "Find $3^{\\frac{1}{3}} \\cdot 9^{\\frac{1}{9}} \\cdot 27^{\\frac{1}{27}} \\cdot 81^{\\frac{1}{81}} \\dotsm.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt[4]{27}",
"style": "rule"
} | {
"index": "38946",
"split": "train"
} |
math | [
{
"content": "Given that $a-b=5$ and $a^2+b^2=35$, find $a^3-b^3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "38947",
"split": "train"
} |
math | [
{
"content": "In the diagram, points \\(B\\), \\(C\\), and \\(D\\) lie on a line. Also, \\(\\angle ABC = 90^\\circ\\) and \\(\\angle ACD = 150^\\circ\\). The value of \\(x\\) is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "38948",
"split": "train"
} |
math | [
{
"content": "Given that $n$ is a positive integer, and $4^7 + 4^n + 4^{1998}$ is a perfect square, then one value of $n$ is. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3988",
"style": "rule"
} | {
"index": "38949",
"split": "train"
} |
math | [
{
"content": "Determine the number of distinct odd numbers that can be formed by rearranging the digits of the number \"34396\". Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "38950",
"split": "train"
} |
math | [
{
"content": "Let\n\\[x^8 - 98x^4 + 1 = p(x) q(x),\\]where $p(x)$ and $q(x)$ are monic, non-constant polynomials with integer coefficients. Find $p(1) + q(1).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "38951",
"split": "train"
} |
math | [
{
"content": "Let $a_{1}, a_{2}, \\ldots$ be a sequence defined by $a_{1}=a_{2}=1$ and $a_{n+2}=a_{n+1}+a_{n}$ for $n \\geq 1$. Find $$\\sum_{n=1}^{\\infty} \\frac{a_{n}}{4^{n+1}}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{11}",
"style": "rule"
} | {
"index": "38952",
"split": "train"
} |
math | [
{
"content": "The floor plan of a castle wall is a regular pentagon with a side length of \\( a (= 100 \\text{ m}) \\). The castle is patrolled by 3 guards along paths from which every point can see the base shape of the castle wall at angles of \\( 90^\\circ, 60^\\circ, \\) and \\( 54^\\circ \\) respectively. ... | {
"ground_truth": "5.83",
"style": "rule"
} | {
"index": "38953",
"split": "train"
} |
math | [
{
"content": "A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 63. If $m$ is also divisible by 11, what is the greatest possible value of $m$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9702",
"style": "rule"
} | {
"index": "38954",
"split": "train"
} |
math | [
{
"content": "A sports conference has 14 teams in two divisions of 7. How many games are in a complete season for the conference if each team must play every other team in its own division twice and every team in the other division once? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "133",
"style": "rule"
} | {
"index": "38955",
"split": "train"
} |
math | [
{
"content": "The positive integers \\( r \\), \\( s \\), and \\( t \\) have the property that \\( r \\times s \\times t = 1230 \\). What is the smallest possible value of \\( r + s + t \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "38956",
"split": "train"
} |
math | [
{
"content": "Given that $x$, $y$, $z \\in \\mathbb{R}$, if $-1$, $x$, $y$, $z$, $-3$ form a geometric sequence, calculate the value of $xyz$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3\\sqrt{3}",
"style": "rule"
} | {
"index": "38957",
"split": "train"
} |
math | [
{
"content": "Each row of a seating arrangement seats 7 or 8 people. Forty-six people are to be seated. How many rows seat exactly 8 people if every seat is occupied? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "38958",
"split": "train"
} |
math | [
{
"content": "In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$ - and $y$ - axes, and call a set of points $nice$ if no two of them have the same $x$ - or $y$ - coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate pl... | {
"ground_truth": "2015",
"style": "rule"
} | {
"index": "38959",
"split": "train"
} |
math | [
{
"content": "In the convex quadrilateral \\(ABCD\\): \\(AB = AC = AD = BD\\) and \\(\\angle BAC = \\angle CBD\\). Find \\(\\angle ACD\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "38960",
"split": "train"
} |
math | [
{
"content": "Severus Snape, the potions professor, prepared three potions, each in an equal volume of 400 ml. The first potion makes the drinker smarter, the second makes them more beautiful, and the third makes them stronger. To ensure the effect of any potion, it is sufficient to drink at least 30 ml of that... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "38961",
"split": "train"
} |
math | [
{
"content": "The force needed to loosen a bolt varies inversely with the length of the handle of the wrench used. A wrench with a handle length of 9 inches requires 375 pounds of force to loosen a certain bolt. A wrench of 15 inches will require how many pounds of force to loosen the same bolt? Please output t... | {
"ground_truth": "225",
"style": "rule"
} | {
"index": "38962",
"split": "train"
} |
math | [
{
"content": "Four girls and eight boys came for a class photograph. Children approach the photographer in pairs and take a joint photo. Among how many minimum photos must there necessarily be either a photo of two boys, a photo of two girls, or two photos with the same children? Please output the final answer ... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "38963",
"split": "train"
} |
math | [
{
"content": "In July 1861, $366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{366}{31 \\times 24}",
"style": "rule"
} | {
"index": "38964",
"split": "train"
} |
math | [
{
"content": "$44 \\times 22$ is equal to Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$88 \\times 11$",
"style": "rule"
} | {
"index": "38965",
"split": "train"
} |
math | [
{
"content": "If four different positive integers $m$, $n$, $p$, $q$ satisfy $(7-m)(7-n)(7-p)(7-q)=4$, find the value of $m+n+p+q$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "38966",
"split": "train"
} |
math | [
{
"content": "The number of integers \\(N\\) from 1 to 1990 for which \\(\\frac{N^{2}+7}{N+4}\\) is not a reduced fraction is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "86",
"style": "rule"
} | {
"index": "38967",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\left\\lfloor |{-34.1}|\\right\\rfloor$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "38968",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow {a}$ = (4, -7) and $\\overrightarrow {b}$ = (3, -4), find the projection of $\\overrightarrow {a}$ - $2\\overrightarrow {b}$ in the direction of $\\overrightarrow {b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "38969",
"split": "train"
} |
math | [
{
"content": "Say that a polynomial with real coefficients in two variables, $x,y$, is \\emph{balanced} if\nthe average value of the polynomial on each circle centered at the origin is $0$.\nThe balanced polynomials of degree at most $2009$ form a vector space $V$ over $\\mathbb{R}$.\nFind the dimension of $V$.... | {
"ground_truth": "2020050",
"style": "rule"
} | {
"index": "38970",
"split": "train"
} |
math | [
{
"content": "Find the product of all real values of $r$ for which $\\frac{1}{2x}=\\frac{r-x}{7}$ has exactly one real solution. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-14",
"style": "rule"
} | {
"index": "38971",
"split": "train"
} |
math | [
{
"content": "On the board, two sums are written:\n\n\\[1+22+333+4444+55555+666666+7777777+88888888+999999999\\]\n\n\\[9+98+987+9876+98765+987654+9876543+98765432+987654321\\]\n\nDetermine which one is greater (or if they are equal). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1097393685",
"style": "rule"
} | {
"index": "38972",
"split": "train"
} |
math | [
{
"content": "If $x^{2y}= 4$ and $x = 4$, what is the value of $y$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "38973",
"split": "train"
} |
math | [
{
"content": "We have $2022$ $1s$ written on a board in a line. We randomly choose a strictly increasing sequence from ${1, 2, . . . , 2022}$ such that the last term is $2022$ . If the chosen sequence is $a_1, a_2, ..., a_k$ ( $k$ is not fixed), then at the $i^{th}$ step, we choose the first a $_i$ ... | {
"ground_truth": "1012",
"style": "rule"
} | {
"index": "38974",
"split": "train"
} |
math | [
{
"content": "Find the number of six-digit palindromes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9000",
"style": "rule"
} | {
"index": "38975",
"split": "train"
} |
math | [
{
"content": "Compute $1-2+3-4+ \\dots -98+99$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "38976",
"split": "train"
} |
math | [
{
"content": "Given that \\( M \\) is a subset of \\(\\{1, 2, 3, \\cdots, 15\\}\\) such that the product of any 3 distinct elements of \\( M \\) is not a perfect square, determine the maximum possible number of elements in \\( M \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "38977",
"split": "train"
} |
math | [
{
"content": "A square has vertices \\( P, Q, R, S \\) labelled clockwise. An equilateral triangle is constructed with vertices \\( P, T, R \\) labelled clockwise. What is the size of angle \\( \\angle RQT \\) in degrees? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "135",
"style": "rule"
} | {
"index": "38978",
"split": "train"
} |
math | [
{
"content": "Given triangle $ ABC$ of area 1. Let $ BM$ be the perpendicular from $ B$ to the bisector of angle $ C$ . Determine the area of triangle $ AMC$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "38979",
"split": "train"
} |
math | [
{
"content": "Dad says he is exactly 35 years old, not counting weekends. How old is he really? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "38980",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{m}=(\\sin x, -1)$ and $\\overrightarrow{n}=(\\sqrt{3}\\cos x, -\\frac{1}{2})$, let $f(x)=(\\overrightarrow{m}+\\overrightarrow{n})\\cdot \\overrightarrow{m}$.\n(1) Find the analytic expression for $f(x)$ and its intervals of monotonic increase;\n(2) Given that $a$, ... | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "38981",
"split": "train"
} |
math | [
{
"content": "The integers \\(a, b,\\) and \\(c\\) satisfy the equations \\(a + 5 = b\\), \\(5 + b = c\\), and \\(b + c = a\\). What is the value of \\(b\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-10",
"style": "rule"
} | {
"index": "38982",
"split": "train"
} |
math | [
{
"content": "In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake ha... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "38983",
"split": "train"
} |
math | [
{
"content": "$\\frac{2}{1-\\frac{2}{3}}=$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "38984",
"split": "train"
} |
math | [
{
"content": "A relay race preparatory team is composed of members from 8 school basketball teams and 2 school football teams, totaling 10 people. If 2 people are randomly selected, find the probability that, given one of them is a football team member, the other one is also a football team member. Please outpu... | {
"ground_truth": "\\frac{1}{17}",
"style": "rule"
} | {
"index": "38985",
"split": "train"
} |
math | [
{
"content": "Given that $\\frac 1n - \\frac{1}{n+1} < \\frac{1}{10}$, what is the least possible positive integer value of $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "38986",
"split": "train"
} |
math | [
{
"content": "The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle, with the vertices around the circumference in the given order. Given that the polygon $P_1P_3P_5P_7$ is a square of area 5, and the polygon $P_2P_4P_6P_8$ is a rectangle of area 4, find the maximum possible area of the octagon. Plea... | {
"ground_truth": "3\\sqrt{5}",
"style": "rule"
} | {
"index": "38987",
"split": "train"
} |
math | [
{
"content": "The natural number \\(a\\) is divisible by 35 and has 75 different divisors, including 1 and \\(a\\). Find the smallest such \\(a\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "490000",
"style": "rule"
} | {
"index": "38988",
"split": "train"
} |
math | [
{
"content": "Let \\( p, q, \\) and \\( r \\) be the roots of the equation \\( x^3 - 15x^2 + 25x - 10 = 0 \\). Find the value of \\( (1+p)(1+q)(1+r) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "38989",
"split": "train"
} |
math | [
{
"content": "How many ways are there to arrange 5 identical red balls and 5 identical blue balls in a line if there cannot be three or more consecutive blue balls in the arrangement? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "126",
"style": "rule"
} | {
"index": "38990",
"split": "train"
} |
math | [
{
"content": "If $x+2 y-3 z=7$ and $2 x-y+2 z=6$, determine $8 x+y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "38991",
"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": "38992",
"split": "train"
} |
math | [
{
"content": "Let the sequence \\( A = (a_1, a_2, \\dots, a_{110}) \\) be such that its Cesaro sum is 1200. Calculate the Cesaro sum of the 111-term sequence \\( (0, a_1, a_2, \\dots, a_{110}) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{132000}{111}",
"style": "rule"
} | {
"index": "38993",
"split": "train"
} |
math | [
{
"content": "A point $M$ on the parabola $y=4x^{2}$ is at a distance of $1$ from the focus. The ordinate of point $M$ is __________. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{15}{16}",
"style": "rule"
} | {
"index": "38994",
"split": "train"
} |
math | [
{
"content": "What is the length of the diagonal of a square with side length $50\\sqrt{2}$ cm? Express your answer in simplest form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "38995",
"split": "train"
} |
math | [
{
"content": "The product $11 \\cdot 30 \\cdot N$ is an integer whose representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is the fourth power of an integer in decimal (base 10). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "38996",
"split": "train"
} |
math | [
{
"content": "A certain store sells a type of handbag, and it is known that the cost price of this handbag is $50$ yuan per unit. Market research shows that the daily sales quantity $y$ (unit: units) of this handbag is related to the selling price $x$ (unit: yuan) as follows: $y=-x+80$ $(50 < x < 80)$. Let $w$ ... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "38997",
"split": "train"
} |
math | [
{
"content": "In the arithmetic sequence $\\{a_{n}\\}$, $\\frac{a_{1010}}{a_{1009}} < -1$. If its first $n$ terms' sum $S_{n}$ has a maximum value, then the maximum positive integer value of $n$ that makes $S_{n} > 0$ is $\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.$ Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "2018",
"style": "rule"
} | {
"index": "38998",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC,$ we have $\\angle C = 3\\angle A,$ $a = 27,$ and $c = 48.$ What is $b$?\n\nNote: $a$ is the side length opposite $\\angle A,$ etc. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "38999",
"split": "train"
} |
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