data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given 15 points in space, 5 of which are collinear, what is the maximum possible number of unique planes that can be determined by these points? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "445",
"style": "rule"
} | {
"index": "1700",
"split": "train"
} |
math | [
{
"content": "A certain type of beverage with a prize promotion has bottle caps printed with either \"reward one bottle\" or \"thank you for your purchase\". If a bottle is purchased and the cap is printed with \"reward one bottle\", it is considered a winning bottle, with a winning probability of $\\frac{1}{6}... | {
"ground_truth": "\\frac{25}{27}",
"style": "rule"
} | {
"index": "1701",
"split": "train"
} |
math | [
{
"content": "Equilateral $\\triangle ABC$ has side length $2$, $M$ is the midpoint of $\\overline{AC}$, and $C$ is the midpoint of $\\overline{BD}$. What is the area of $\\triangle CDM$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "1702",
"split": "train"
} |
math | [
{
"content": "The measure of angle $ACB$ is 30 degrees. If ray $CA$ is rotated 510 degrees about point $C$ clockwise, what will be the positive measure of the acute angle $ACB$, in degrees? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "1703",
"split": "train"
} |
math | [
{
"content": "Let $z$ be a complex number such that\n\\[z^2 + |z|^2 = 3 - 5i.\\]Find $|z|^2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17}{3}",
"style": "rule"
} | {
"index": "1704",
"split": "train"
} |
math | [
{
"content": "120 granite slabs weighing 7 tons each and 80 slabs weighing 9 tons each have been stockpiled at the quarry. A railroad platform can hold up to 40 tons. What is the minimum number of platforms required to transport all the slabs? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "1705",
"split": "train"
} |
math | [
{
"content": "In the acute-angled triangle \\(ABC\\), it is known that \\(\\sin (A+B)=\\frac{3}{5}\\), \\(\\sin (A-B)=\\frac{1}{5}\\), and \\(AB=3\\). Find the area of \\(\\triangle ABC\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{6 + 3\\sqrt{6}}{2}",
"style": "rule"
} | {
"index": "1706",
"split": "train"
} |
math | [
{
"content": "Solve for $y$: $3y+7y = 282-8(y-3)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "1707",
"split": "train"
} |
math | [
{
"content": "A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? Please output the final answer within \\bo... | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "1708",
"split": "train"
} |
math | [
{
"content": "In general, if there are $d$ doors in every room (but still only 1 correct door) and $r$ rooms, the last of which leads into Bowser's level, what is the expected number of doors through which Mario will pass before he reaches Bowser's level? Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "\\frac{d\\left(d^{r}-1\\right)}{d-1}",
"style": "rule"
} | {
"index": "1709",
"split": "train"
} |
math | [
{
"content": "Bob's favorite number is between $50$ and $100$. It is a multiple of $11$, but not a multiple of $2$. The sum of its digits is a multiple of $3$. What is Bob's favorite number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "99",
"style": "rule"
} | {
"index": "1710",
"split": "train"
} |
math | [
{
"content": "The states of Sunshine and Prairie have adopted new license plate configurations. Sunshine license plates are formatted \"LDDLDL\" where L denotes a letter and D denotes a digit. Prairie license plates are formatted \"LDDLDD\". Assuming all 10 digits and 26 letters are equally likely to appear in ... | {
"ground_truth": "10816000",
"style": "rule"
} | {
"index": "1711",
"split": "train"
} |
math | [
{
"content": "For how many integers \\( n \\) is \\(\\frac{2n^3 - 12n^2 - 2n + 12}{n^2 + 5n - 6}\\) equal to an integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "1712",
"split": "train"
} |
math | [
{
"content": "How many different five-letter words can be formed such that they start and end with the same letter, and the middle letter is always 'A'? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17576",
"style": "rule"
} | {
"index": "1713",
"split": "train"
} |
math | [
{
"content": "Place 5 balls, numbered 1, 2, 3, 4, 5, into three different boxes, with two boxes each containing 2 balls and the other box containing 1 ball. How many different arrangements are there? (Answer with a number). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "1714",
"split": "train"
} |
math | [
{
"content": "If $\\log_6 x=2.5$, the value of $x$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36\\sqrt{6}",
"style": "rule"
} | {
"index": "1715",
"split": "train"
} |
math | [
{
"content": "The points $P(3,-2), Q(3,1), R(7,1)$, and $S$ form a rectangle. What are the coordinates of $S$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(7,-2)",
"style": "rule"
} | {
"index": "1716",
"split": "train"
} |
math | [
{
"content": "Mia is \"helping\" her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her ... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "1717",
"split": "train"
} |
math | [
{
"content": "Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise.\nIn a turn of the game, Alice moves 5 points clockwise and Bob moves 9 ... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "1718",
"split": "train"
} |
math | [
{
"content": "Let $d_1 = a^2 + 2^a + a \\cdot 2^{(a+1)/2}$ and $d_2 = a^2 + 2^a - a \\cdot 2^{(a+1)/2}$. If $1 \\le a \\le 251$, how many integral values of $a$ are there such that $d_1 \\cdot d_2$ is a multiple of $5$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "1719",
"split": "train"
} |
math | [
{
"content": "An ellipse has a focus at coordinates $\\left(0,-\\sqrt {2}\\right)$ and is represented by the equation $2x^{2}-my^{2}=1$. Find the value of the real number $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\dfrac{2}{5}",
"style": "rule"
} | {
"index": "1720",
"split": "train"
} |
math | [
{
"content": "Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\\triangle ABC$ be $a$, $b$, $c$, respectively, and $C=\\frac{π}{3}$, $c=2$. Then find the maximum value of $\\overrightarrow{AC}•\\overrightarrow{AB}$. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "\\frac{4\\sqrt{3}}{3} + 2",
"style": "rule"
} | {
"index": "1721",
"split": "train"
} |
math | [
{
"content": "The equation of curve $C$ is $\\frac{x^2}{m^2} + \\frac{y^2}{n^2} = 1$, where $m$ and $n$ are the numbers obtained by rolling a die twice in succession. Let event $A$ be \"The equation $\\frac{x^2}{m^2} + \\frac{y^2}{n^2} = 1$ represents an ellipse with foci on the $x$-axis\". Then, the probabilit... | {
"ground_truth": "\\frac{5}{12}",
"style": "rule"
} | {
"index": "1722",
"split": "train"
} |
math | [
{
"content": "At a sumo wrestling tournament, 20 sumo wrestlers participated. After weighing, it was found that the average weight of the wrestlers is 125 kg. What is the maximum possible number of wrestlers weighing more than 131 kg, given that according to the rules, individuals weighing less than 90 kg canno... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "1723",
"split": "train"
} |
math | [
{
"content": "Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \\[x^3 - 10x^2 + px - q = 0\\] are three distinct positive integers. Compute $p + q.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "1724",
"split": "train"
} |
math | [
{
"content": "Abe, Bobby, Charles, Devin and Edwin are the participants in a race. How many different 1st-2nd-3rd place outcomes are possible if there are no ties? Two different outcomes to include are Bobby-Devin-Edwin and Devin-Bobby-Edwin. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "1725",
"split": "train"
} |
math | [
{
"content": "If $\\frac{1}{x} + \\frac{1}{y} = 3$ and $\\frac{1}{x} - \\frac{1}{y} = -7$ what is the value of $x + y$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{10}",
"style": "rule"
} | {
"index": "1726",
"split": "train"
} |
math | [
{
"content": "In base $b$, there are exactly one hundred three-digit numbers whose digits are all distinct. (That's \"one hundred\" in the ordinary sense, $100_{10}$.)\n\nWhat is $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "1727",
"split": "train"
} |
math | [
{
"content": "Find $x$, given that $x$ is neither zero nor one and the numbers $\\{x\\}$, $\\lfloor x \\rfloor$, and $x$ form a geometric sequence in that order. (Recall that $\\{x\\} = x - \\lfloor x\\rfloor$). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.618",
"style": "rule"
} | {
"index": "1728",
"split": "train"
} |
math | [
{
"content": "Suppose the roots of the polynomial $Q(x) = x^3 + px^2 + qx + r$ are $\\sin \\frac{3\\pi}{7}, \\sin \\frac{5\\pi}{7},$ and $\\sin \\frac{\\pi}{7}$. Calculate the product $pqr$.\nA) $0.725$\nB) $1.45$\nC) $1.0$\nD) $1.725$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.725",
"style": "rule"
} | {
"index": "1729",
"split": "train"
} |
math | [
{
"content": "If $a\\ast b = 3a+4b-ab$, what is the value of $5\\ast2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "1730",
"split": "train"
} |
math | [
{
"content": "Find the smallest possible side of a square in which five circles of radius $1$ can be placed, so that no two of them have a common interior point. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 + 2\\sqrt{2}",
"style": "rule"
} | {
"index": "1731",
"split": "train"
} |
math | [
{
"content": "Find the value of $a$ that satisfies the equation $293_{a}+468_{a}=73B_{a}$, where $B_{a}=11_{10}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "1732",
"split": "train"
} |
math | [
{
"content": "For how many positive integers $x$ is $x^2 + 6x + 9$ between 20 and 40? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "1733",
"split": "train"
} |
math | [
{
"content": "A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder cuts off a truncated cone with a volume of 91 from the original cone. Find the volume of the original cone. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "94.5",
"style": "rule"
} | {
"index": "1734",
"split": "train"
} |
math | [
{
"content": "What is the least positive multiple of 25 for which the product of its digits is also a positive multiple of 25? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "525",
"style": "rule"
} | {
"index": "1735",
"split": "train"
} |
math | [
{
"content": "There are three pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ that satisfy both $x^3-3xy^2=2005$ and $y^3-3x^2y=2004$. Compute $\\left(1-\\frac{x_1}{y_1}\\right)\\left(1-\\frac{x_2}{y_2}\\right)\\left(1-\\frac{x_3}{y_3}\\right)$. Please output the final answer within \\boxed{}.",... | {
"ground_truth": "\\frac{1}{1002}",
"style": "rule"
} | {
"index": "1736",
"split": "train"
} |
math | [
{
"content": "Given that $\\alpha$ is an angle in the second quadrant, simplify $$\\frac { \\sqrt {1+2\\sin(5\\pi-\\alpha)\\cos(\\alpha-\\pi)}}{\\sin\\left(\\alpha - \\frac {3}{2}\\pi \\right)- \\sqrt {1-\\sin^{2}\\left( \\frac {3}{2}\\pi+\\alpha\\right)}}.$$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "1737",
"split": "train"
} |
math | [
{
"content": "The number of solutions to \\{1,~2\\} \\subseteq~X~\\subseteq~\\{1,~2,~3,~4,~5\\}, where $X$ is a subset of \\{1,~2,~3,~4,~5\\} is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "1738",
"split": "train"
} |
math | [
{
"content": "A function $f(x)$ defined on $R$ satisfies $f(x+1) = 2f(x)$. When $x \\in (-1,0]$, $f(x) = x^{3}$. Find $f(\\frac{21}{2})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-256",
"style": "rule"
} | {
"index": "1739",
"split": "train"
} |
math | [
{
"content": "Two rectangles have integer dimensions, and both have a perimeter of 144 cm. What is the greatest possible difference between the areas of two such rectangles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1225",
"style": "rule"
} | {
"index": "1740",
"split": "train"
} |
math | [
{
"content": "If $3x-5=10x+9$, what is $4(x+7)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "1741",
"split": "train"
} |
math | [
{
"content": "Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\\angle ABD$ exceeds $\\angle AHG$ by $12^\\circ$. Find the degree measure of $\\angle BA... | {
"ground_truth": "58",
"style": "rule"
} | {
"index": "1742",
"split": "train"
} |
math | [
{
"content": "An $m\\times n\\times p$ rectangular box has half the volume of an $(m + 2)\\times(n + 2)\\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\\le n\\le p.$ What is the largest possible value of $p$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "130",
"style": "rule"
} | {
"index": "1743",
"split": "train"
} |
math | [
{
"content": "Given a set of data is multiplied by 2 and then reduced by 80 for each data point, resulting in a new set of data with an average of 1.2 and a variance of 4.4, determine the average and variance of the original data. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.1",
"style": "rule"
} | {
"index": "1744",
"split": "train"
} |
math | [
{
"content": "The number of triangles with all sides being positive integers and the longest side being 11 can be expressed as a combination problem, calculate the number of such triangles. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "1745",
"split": "train"
} |
math | [
{
"content": "Given integers $a$ and $b$ whose sum is $20$, determine the number of distinct integers that can be written as the sum of two, not necessarily different, fractions $\\frac{a}{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "1746",
"split": "train"
} |
math | [
{
"content": "A $4\\times 4\\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$? Please output the final answer within \\b... | {
"ground_truth": "$2+2\\sqrt 7$",
"style": "rule"
} | {
"index": "1747",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=4-x^{2}+a\\ln x$, if $f(x)\\leqslant 3$ for all $x > 0$, determine the range of the real number $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "[2]",
"style": "rule"
} | {
"index": "1748",
"split": "train"
} |
math | [
{
"content": "If $991+993+995+997+999=5000-N$, then $N=$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "1749",
"split": "train"
} |
math | [
{
"content": "For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$ . $z$ is the maximum real number such that [ $\\frac{5}{z}$ ] + [ $\\frac{6}{z}$ ] = 7. Find the value of $ 20z$ . Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "1750",
"split": "train"
} |
math | [
{
"content": "Find the common ratio of the infinite geometric series: $$\\frac{5}{6}-\\frac{4}{9}+\\frac{32}{135}-\\dots$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{8}{15}",
"style": "rule"
} | {
"index": "1751",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 420 = 0$ has integral solutions? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "130",
"style": "rule"
} | {
"index": "1752",
"split": "train"
} |
math | [
{
"content": "How many of the smallest 216 positive integers written in base 6 use the digit 3 or 4 (or both) as a digit? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "168",
"style": "rule"
} | {
"index": "1753",
"split": "train"
} |
math | [
{
"content": "Given that the focus of the parabola $y^{2}=ax$ coincides with the left focus of the ellipse $\\frac{x^{2}}{6}+ \\frac{y^{2}}{2}=1$, find the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-16",
"style": "rule"
} | {
"index": "1754",
"split": "train"
} |
math | [
{
"content": "Let $T$ be the set of ordered triples $(x,y,z)$, where $x,y,z$ are integers with $0\\leq x,y,z\\leq9$. Players $A$ and $B$ play the following guessing game. Player $A$ chooses a triple $(x,y,z)$ in $T$, and Player $B$ has to discover $A$[i]'s[/i] triple in as few moves as possible. A [i]move[/i] ... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "1755",
"split": "train"
} |
math | [
{
"content": "Find the area of the ellipse given by $x^2 + 6x + 4y^2 - 8y + 9 = 0.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\pi",
"style": "rule"
} | {
"index": "1756",
"split": "train"
} |
math | [
{
"content": "How many of the positive divisors of 3960 are multiples of 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "1757",
"split": "train"
} |
math | [
{
"content": "In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a $1$ on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins $3$ games before Jacob wins ... | {
"ground_truth": "360",
"style": "rule"
} | {
"index": "1758",
"split": "train"
} |
math | [
{
"content": "Find $x$ if $\\log_9(2x-7) = \\dfrac{3}{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "1759",
"split": "train"
} |
math | [
{
"content": "On the first day, Barry Sotter used his magic to make an object's length increase by $\\frac{1}{3}$, so if the original length was $x$, it became $x + \\frac{1}{3} x$. On the second day, he increased the new length by $\\frac{1}{4}$; on the third day by $\\frac{1}{5}$; and so on. On the $n^{\\text... | {
"ground_truth": "147",
"style": "rule"
} | {
"index": "1760",
"split": "train"
} |
math | [
{
"content": "For any two non-zero plane vectors $\\overrightarrow \\alpha$ and $\\overrightarrow \\beta$, a new operation $\\odot$ is defined as $\\overrightarrow \\alpha ⊙ \\overrightarrow \\beta = \\frac{\\overrightarrow \\alpha • \\overrightarrow \\beta}{\\overrightarrow \\beta • \\overrightarrow \\beta}$. ... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "1761",
"split": "train"
} |
math | [
{
"content": "For point M on the ellipse $\\frac {x^{2}}{9}+ \\frac {y^{2}}{4}=1$, find the minimum distance from M to the line $x+2y-10=0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "1762",
"split": "train"
} |
math | [
{
"content": "Let $A$ and $B$ be two points on the parabola $y = x^2,$ such that when the tangents at $A$ and $B$ drawn, they are perpendicular. Then for any such pair of tangents, the $y$-coordinate of their point of intersection $P$ is always the same. Find this $y$-coordinate.\n\n[asy]\nunitsize(1.5 cm);\n... | {
"ground_truth": "-\\frac{1}{4}",
"style": "rule"
} | {
"index": "1763",
"split": "train"
} |
math | [
{
"content": "Let \\( f \\) be a function on \\([0,1]\\) satisfying:\n1. \\( f(0) = 0, f(1) = 1 \\).\n2. For all \\( x, y \\in [0, 1] \\), with \\( x \\leq y \\), \\( f\\left(\\frac{x+y}{2}\\right) = (1-a) f(x) + a f(y) \\), where \\( a \\) is a real number and \\( 0 \\leq a \\leq 1 \\). Find \\( f\\left(\\frac... | {
"ground_truth": "\\frac{1}{7}",
"style": "rule"
} | {
"index": "1764",
"split": "train"
} |
math | [
{
"content": "Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\\subseteq\\{1,2,\\ldots,2n\\}$ , for integers $i,j$ with $0\\leq i,j\\leq n$ , such that:\n$\\bullet$ for all $0\\leq i,j\\leq n$ , the set $S_{i,j}$ has $i+j$ elements; and\n$\\bullet$ $S_{... | {
"ground_truth": "\\[\n(2n)! \\cdot 2^{n^2}\n\\]",
"style": "rule"
} | {
"index": "1765",
"split": "train"
} |
math | [
{
"content": "Add $101_2 + 11_2 + 1100_2 + 11101_2.$ Express your answer in base $2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "110001_2",
"style": "rule"
} | {
"index": "1766",
"split": "train"
} |
math | [
{
"content": "Analogous to the exponentiation of rational numbers, we define the division operation of several identical rational numbers (all not equal to $0$) as \"division exponentiation,\" denoted as $a^{ⓝ}$, read as \"$a$ circle $n$ times.\" For example, $2\\div 2\\div 2$ is denoted as $2^{③}$, read as \"$... | {
"ground_truth": "-73",
"style": "rule"
} | {
"index": "1767",
"split": "train"
} |
math | [
{
"content": "In the number $56439.2071$, the value of the place occupied by the digit 6 is how many times as great as the value of the place occupied by the digit 2? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10,000",
"style": "rule"
} | {
"index": "1768",
"split": "train"
} |
math | [
{
"content": "A building has three different staircases, all starting at the base of the building and ending at the top. One staircase has 104 steps, another has 117 steps, and the other has 156 steps. Whenever the steps of the three staircases are at the same height, there is a floor. How many floors does the ... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "1769",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $C_{1}: \\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ and the hyperbola $C_{2}: \\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1$ with asymptote equations $x \\pm \\sqrt{3}y = 0$, determine the product of the eccentricities of $C_{1}$ and $C_{2}$. Please output t... | {
"ground_truth": "\\frac{2\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "1770",
"split": "train"
} |
math | [
{
"content": "We have two concentric circles $C_{1}$ and $C_{2}$ with radii 1 and 2, respectively. A random chord of $C_{2}$ is chosen. What is the probability that it intersects $C_{1}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "N/A",
"style": "rule"
} | {
"index": "1771",
"split": "train"
} |
math | [
{
"content": "A certain product was bought in the fall, and 825 rubles were paid for it. In the fall, the price per kilogram of this product was 1 ruble cheaper than in the spring. Therefore, for the same amount in the spring, 220 kg less was bought. How much does 1 kg of the product cost in the spring, and how... | {
"ground_truth": "550",
"style": "rule"
} | {
"index": "1772",
"split": "train"
} |
math | [
{
"content": "Let $\\log_{4}3=x$. Then $\\log_{2}27=kx$. Find $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "k=6",
"style": "rule"
} | {
"index": "1773",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is known that $\\overrightarrow{a}=(\\cos A,\\cos B)$, $\\overrightarrow{b}=(a,2c-b)$, and $\\overrightarrow{a} \\parallel \\overrightarrow{b}$.\n(Ⅰ) Find the magnitude of angle $A$;\n(Ⅱ) If ... | {
"ground_truth": "\\sqrt {13}",
"style": "rule"
} | {
"index": "1774",
"split": "train"
} |
math | [
{
"content": "The function $\\mathbf{y}=f(x)$ satisfies the following conditions:\n\na) $f(4)=2$;\n\nb) $f(n+1)=\\frac{1}{f(0)+f(1)}+\\frac{1}{f(1)+f(2)}+\\ldots+\\frac{1}{f(n)+f(n+1)}, n \\geq 0$.\n\nFind the value of $f(2022)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2022}",
"style": "rule"
} | {
"index": "1775",
"split": "train"
} |
math | [
{
"content": "Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang $7$ songs, which was more than any other girl, and Mary sang $4$ songs, which was fewer than any other girl. How many songs did these trios sing? Please output the final an... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "1776",
"split": "train"
} |
math | [
{
"content": "In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10}{3}",
"style": "rule"
} | {
"index": "1777",
"split": "train"
} |
math | [
{
"content": "If the arithmetic mean of two numbers is $6$ and their geometric mean is $10$, then an equation with the given two numbers as roots is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "x^2 - 12x + 100 = 0",
"style": "rule"
} | {
"index": "1778",
"split": "train"
} |
math | [
{
"content": "Find the number of units in the length of diagonal $DA$ of the regular hexagon shown. Express your answer in simplest radical form. [asy]\nsize(120);\ndraw((1,0)--(3,0)--(4,1.732)--(3,3.464)--(1,3.464)--(0,1.732)--cycle);\ndraw((1,0)--(1,3.464));\nlabel(\"10\",(3.5,2.598),NE);\nlabel(\"$A$\",(1,0)... | {
"ground_truth": "10\\sqrt{3}",
"style": "rule"
} | {
"index": "1779",
"split": "train"
} |
math | [
{
"content": "A teacher has a class with $24$ students in it. If she wants to split the students into equal groups of at most $10$ students each, what is the least number of groups that she needs? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "1780",
"split": "train"
} |
math | [
{
"content": "The volume of a sphere is increased to $72\\pi$ cubic inches. What is the new surface area of the sphere? Express your answer in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36\\pi \\cdot 2^{2/3}",
"style": "rule"
} | {
"index": "1781",
"split": "train"
} |
math | [
{
"content": "The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.\n\n[asy]\nunitsize(0.8cm);\nfor (int i=0; i<7; ++i) {\nfor (int j=0; j<4; ++j) {\ndot((i,j));\n};}\nlabel(\"$A$\",(0,3),W);\nlabel(\"$B$\",(6,0),E... | {
"ground_truth": "\\frac{5\\sqrt{5}}{3}",
"style": "rule"
} | {
"index": "1782",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = x^2 - 3x - 1$, find the derivative of $f(2)$ and $f'(1)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "1783",
"split": "train"
} |
math | [
{
"content": "Two distinct natural numbers end in 5 zeros and have exactly 42 divisors. Find their sum. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "700000",
"style": "rule"
} | {
"index": "1784",
"split": "train"
} |
math | [
{
"content": "\\( z_{1}, z_{2}, z_{3} \\) are the three roots of the polynomial\n\\[ P(z) = z^{3} + a z + b \\]\nand satisfy the condition\n\\[ \\left|z_{1}\\right|^{2} + \\left|z_{2}\\right|^{2} + \\left|z_{3}\\right|^{2} = 250 \\]\nMoreover, the three points \\( z_{1}, z_{2}, z_{3} \\) in the complex plane fo... | {
"ground_truth": "5\\sqrt{15}",
"style": "rule"
} | {
"index": "1785",
"split": "train"
} |
math | [
{
"content": "The numbers \\( 62, 63, 64, 65, 66, 67, 68, 69, \\) and \\( 70 \\) are divided by, in some order, the numbers \\( 1, 2, 3, 4, 5, 6, 7, 8, \\) and \\( 9 \\), resulting in nine integers. The sum of these nine integers is \\( S \\). What are the possible values of \\( S \\)? Please output the final a... | {
"ground_truth": "187",
"style": "rule"
} | {
"index": "1786",
"split": "train"
} |
math | [
{
"content": "Given sets \\( A = \\{ x \\mid 5x - a \\leq 0 \\} \\) and \\( B = \\{ x \\mid 6x - b > 0 \\} \\), where \\( a, b \\in \\mathbf{N} \\), and \\( A \\cap B \\cap \\mathbf{N} = \\{ 2, 3, 4 \\} \\), the number of integer pairs \\((a, b)\\) is: Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "1787",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "738",
"style": "rule"
} | {
"index": "1788",
"split": "train"
} |
math | [
{
"content": "A gumball machine contains $9$ red, $7$ white, and $8$ blue gumballs. The least number of gumballs a person must buy to be sure of getting four gumballs of the same color is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "1789",
"split": "train"
} |
math | [
{
"content": "Suppose that we are given 52 points equally spaced around the perimeter of a rectangle, such that four of them are located at the vertices, 13 points on two opposite sides, and 12 points on the other two opposite sides. If \\( P \\), \\( Q \\), and \\( R \\) are chosen to be any three of these poi... | {
"ground_truth": "805",
"style": "rule"
} | {
"index": "1790",
"split": "train"
} |
math | [
{
"content": "Let $0 \\le a,$ $b,$ $c \\le 1.$ Find the maximum value of\n\\[\\sqrt{abc} + \\sqrt{(1 - a)(1 - b)(1 - c)}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "1791",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system, the equation of circle $C$ is $(x-2)^2 + y^2 = 4$, with the center at point $C$. Using the origin as the pole and the non-negative half of the $x$-axis as the initial ray, establish a polar coordinate system. The curve $C_1: \\rho = -4\\sqrt{3}\\sin \\theta$ int... | {
"ground_truth": "1 : 2",
"style": "rule"
} | {
"index": "1792",
"split": "train"
} |
math | [
{
"content": "Assume that a certain BASIC language limits identifiers to a maximum of three characters. The first character must be one of the 26 English letters, and the 2nd and 3rd characters can be either English letters or Arabic numerals (0, 1, 2, 3, 4, ..., 9). Find the total number of possible identifier... | {
"ground_truth": "34658",
"style": "rule"
} | {
"index": "1793",
"split": "train"
} |
math | [
{
"content": "Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference of the numbers on the cards is 11, and their product is divisible by 5?\n\nThe order of the selected cards does not matter: for example, selecting cards with numbers 5 and 16, as well as... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "1794",
"split": "train"
} |
math | [
{
"content": "Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\n\\[n = f_1\\cdot f_2\\cdots f_k,\\]where $k\\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the or... | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "1795",
"split": "train"
} |
math | [
{
"content": "Let \\( OP \\) be the diameter of the circle \\( \\Omega \\), and let \\( \\omega \\) be a circle with center at point \\( P \\) and a radius smaller than that of \\( \\Omega \\). The circles \\( \\Omega \\) and \\( \\omega \\) intersect at points \\( C \\) and \\( D \\). A chord \\( OB \\) of the... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "1796",
"split": "train"
} |
math | [
{
"content": "Find the slope of the line $3x+5y=20$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{5}",
"style": "rule"
} | {
"index": "1797",
"split": "train"
} |
math | [
{
"content": "You are given a set of cards labeled from 1 to 100. You wish to make piles of three cards such that in any pile, the number on one of the cards is the product of the numbers on the other two cards. However, no card can be in more than one pile. What is the maximum number of piles you can form at o... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "1798",
"split": "train"
} |
math | [
{
"content": "Let $h(x) = x - 3$ and $k(x) = 2x$. Also denote the inverses of these functions as $h^{-1}$ and $k^{-1}$. Compute:\n\\[ h(k^{-1}(h^{-1}(h^{-1}(k(h(28)))))) \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "1799",
"split": "train"
} |
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