data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given the ellipse $\\frac {x^{2}}{9} + \\frac {y^{2}}{4} = 1$, and the line $L: x + 2y - 10 = 0$.\n\n(1) Does there exist a point $M$ on the ellipse for which the distance to line $L$ is minimal? If so, find the coordinates of point $M$ and the minimum distance.\n\n(2) Does there exist a point $P$... | {
"ground_truth": "3\\sqrt {5}",
"style": "rule"
} | {
"index": "8100",
"split": "train"
} |
math | [
{
"content": "For any positive integer \\( n \\), let \\( f(n) = 70 + n^2 \\) and let \\( g(n) \\) be the greatest common divisor (GCD) of \\( f(n) \\) and \\( f(n+1) \\). Find the greatest possible value of \\( g(n) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "281",
"style": "rule"
} | {
"index": "8101",
"split": "train"
} |
math | [
{
"content": "Simplify and then evaluate the expression:\n$$( \\frac {x}{x-1}- \\frac {x}{x^{2}-1})÷ \\frac {x^{2}-x}{x^{2}-2x+1}$$\nwhere $$x= \\sqrt {2}-1$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1- \\frac { \\sqrt {2}}{2}",
"style": "rule"
} | {
"index": "8102",
"split": "train"
} |
math | [
{
"content": "Let $x$ be chosen at random from the interval $(0,1)$. What is the probability that $\\lfloor\\log_{10}4x\\rfloor - \\lfloor\\log_{10}x\\rfloor = 0$? Here $\\lfloor x\\rfloor$ denotes the greatest integer that is less than or equal to $x$. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "8103",
"split": "train"
} |
math | [
{
"content": "Lori makes a list of all the numbers between $1$ and $999$ inclusive. She first colors all the multiples of $5$ red. Then she colors blue every number which is adjacent to a red number. How many numbers in her list are left uncolored? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "402",
"style": "rule"
} | {
"index": "8104",
"split": "train"
} |
math | [
{
"content": "Cory has $4$ apples, $2$ oranges, and $1$ banana. If Cory eats one piece of fruit per day for a week, and must consume at least one apple before any orange, how many different orders can Cory eat these fruits? The fruits within each category are indistinguishable. Please output the final answer wi... | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "8105",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of the function \\( f(x)=\\cos 4x + 6\\cos 3x + 17\\cos 2x + 30\\cos x \\) for \\( x \\in \\mathbb{R} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-18",
"style": "rule"
} | {
"index": "8106",
"split": "train"
} |
math | [
{
"content": "Let \\(p\\) be a prime number and \\(\\mathbb{F}_{p}\\) be the field of residues modulo \\(p\\). Let \\(W\\) be the smallest set of polynomials with coefficients in \\(\\mathbb{F}_{p}\\) such that the polynomials \\(x+1\\) and \\(x^{p-2}+x^{p-3}+\\cdots+x^{2}+2x+1\\) are in \\(W\\), and for any po... | {
"ground_truth": "p!",
"style": "rule"
} | {
"index": "8107",
"split": "train"
} |
math | [
{
"content": "We roll five dice, each a different color. In how many ways can the sum of the rolls be 11? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "205",
"style": "rule"
} | {
"index": "8108",
"split": "train"
} |
math | [
{
"content": "For how many four-digit whole numbers does the sum of the digits equal $30$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "8109",
"split": "train"
} |
math | [
{
"content": "Let \\\\(f(x)=a(x-5)^{2}+6\\ln x\\\\), where \\\\(a\\in\\mathbb{R}\\\\), the tangent line of the curve \\\\(y=f(x)\\\\) at point \\\\((1,f(1))\\\\) intersects the \\\\(y\\\\)-axis at point \\\\((0,6)\\\\). \n\\\\((1)\\\\) Determine the value of \\\\(a\\\\); \n\\\\((2)\\\\) Find the intervals of ... | {
"ground_truth": "2+6\\ln 3",
"style": "rule"
} | {
"index": "8110",
"split": "train"
} |
math | [
{
"content": "The altitude \\(AH\\) and the angle bisector \\(CL\\) of triangle \\(ABC\\) intersect at point \\(O\\). Find the angle \\(BAC\\) if it is known that the difference between the angle \\(COH\\) and half of the angle \\(ABC\\) is \\(46^\\circ\\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "92",
"style": "rule"
} | {
"index": "8111",
"split": "train"
} |
math | [
{
"content": "A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.\n[asy] import three; import solids; size(5c... | {
"ground_truth": "384",
"style": "rule"
} | {
"index": "8112",
"split": "train"
} |
math | [
{
"content": "In a triangle \\( \\triangle ABC \\), \\(a\\), \\(b\\), and \\(c\\) are the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) respectively, with \\(B= \\dfrac {2\\pi}{3}\\). If \\(a^{2}+c^{2}=4ac\\), then find the value of \\( \\dfrac {\\sin (A+C)}{\\sin A\\sin C} \\). Please output the final... | {
"ground_truth": "\\dfrac{10\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "8113",
"split": "train"
} |
math | [
{
"content": "One of the mascots for the 2012 Olympic Games is called 'Wenlock' because the town of Wenlock in Shropshire first held the Wenlock Olympian Games in 1850. How many years ago was that?\n\nA) 62\n\nB) 152\n\nC) 158\n\nD) 162\n\nE) 172 Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "162",
"style": "rule"
} | {
"index": "8114",
"split": "train"
} |
math | [
{
"content": "Suppose that \\(\\begin{array}{c} a \\\\ b \\\\ c \\end{array}\\) means $a+b-c$.\nFor example, \\(\\begin{array}{c} 5 \\\\ 4 \\\\ 6 \\end{array}\\) is $5+4-6 = 3$.\nThen the sum \\(\\begin{array}{c} 3 \\\\ 2 \\\\ 5 \\end{array}\\) + \\(\\begin{array}{c} 4 \\\\ 1 \\\\ 6 \\end{array}\\) is Please ou... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "8115",
"split": "train"
} |
math | [
{
"content": "When a student multiplied the number $66$ by the repeating decimal, \\(1.\\overline{ab}\\), where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab$. Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit number $ab$? P... | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "8116",
"split": "train"
} |
math | [
{
"content": "The perimeter of triangle \\(ABC\\) is 1. A circle \\(\\omega\\) touches side \\(BC\\), the extension of side \\(AB\\) at point \\(P\\), and the extension of side \\(AC\\) at point \\(Q\\). A line passing through the midpoints of \\(AB\\) and \\(AC\\) intersects the circumcircle of triangle \\(APQ... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "8117",
"split": "train"
} |
math | [
{
"content": "What is the sum of all four-digit integers from 1000 to 5000, where each number increases by 4 from the previous one? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3003000",
"style": "rule"
} | {
"index": "8118",
"split": "train"
} |
math | [
{
"content": "In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $v$, $w$, $x$, $y$, and $z$. Find $y+z$.\n\n[asy]\npath a=(0,0)--(1,0)--(1,1)--(0,1)--cycle;\nfor (int i=0; i<3; ++i) {\nfor (int j=0; j<3; ++j) {\ndraw(shift... | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "8119",
"split": "train"
} |
math | [
{
"content": "Find $x$ such that $\\log_x 81=\\log_2 16$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "8120",
"split": "train"
} |
math | [
{
"content": "A belt is placed without slack around two non-crossing circular pulleys which have radii of $15$ inches and $5$ inches respectively. The distance between the points where the belt contacts the pulleys is $30$ inches. Determine the distance between the centers of the two pulleys.\nA) $20$ inches\nB... | {
"ground_truth": "10\\sqrt{10}",
"style": "rule"
} | {
"index": "8121",
"split": "train"
} |
math | [
{
"content": "The solutions to the equation $(z-4)^6 = 64$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labelled $D, E,$ and $F$. What is the least possible area of triangle $DEF$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "8122",
"split": "train"
} |
math | [
{
"content": "A cuboid has an integer volume. Three of the faces have different areas, namely $7, 27$ , and $L$ . What is the smallest possible integer value for $L$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "8123",
"split": "train"
} |
math | [
{
"content": "Given a positive integer \\( n \\) (\\( n \\geqslant 6 \\)), let set \\( A \\) be composed of sums of any 5 consecutive positive integers not greater than \\( n \\), and let set \\( B \\) be composed of sums of any 6 consecutive positive integers not greater than \\( n \\). If the number of elemen... | {
"ground_truth": "12106",
"style": "rule"
} | {
"index": "8124",
"split": "train"
} |
math | [
{
"content": "Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ respectively in $\\triangle ABC$, and the altitude on side $BC$ is $\\frac{a}{2}$. Determine the maximum value of $\\frac{c}{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2} + 1",
"style": "rule"
} | {
"index": "8125",
"split": "train"
} |
math | [
{
"content": "Convert the binary number $111011001001_{(2)}$ to its corresponding decimal number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3785",
"style": "rule"
} | {
"index": "8126",
"split": "train"
} |
math | [
{
"content": "A prize fund is divided into first, second, and third prizes. The prize for each first prize is 3 times that of each second prize, and the prize for each second prize is 3 times that of each third prize. The total prize fund is 10,800 yuan. If the total prize money for the third prize is more than... | {
"ground_truth": "2700",
"style": "rule"
} | {
"index": "8127",
"split": "train"
} |
math | [
{
"content": "If four distinct positive integers $a$, $b$, $c$, $d$ satisfy $\\left(6-a\\right)\\left(6-b\\right)\\left(6-c\\right)\\left(6-d\\right)=9$, then $a+b+c+d$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "8128",
"split": "train"
} |
math | [
{
"content": "The number of minutes in a week is closest to: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10000",
"style": "rule"
} | {
"index": "8129",
"split": "train"
} |
math | [
{
"content": "Given that a rectangle with length $3x$ inches and width $x + 5$ inches has the property that its area and perimeter have equal values, what is $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "8130",
"split": "train"
} |
math | [
{
"content": "A mathematical contest had $3$ problems, each of which was given a score between $0$ and $7$ ( $0$ and $7$ included). It is known that, for any two contestants, there exists at most one problem in which they have obtained the same score (for example, there are no two contestants whose ord... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "8131",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer with exactly $12$ positive factors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "8132",
"split": "train"
} |
math | [
{
"content": "A list of integers has mode 32 and mean 22. The smallest number in the list is 10. The median m of the list is a member of the list. If the list member m were replaced by m+10, the mean and median of the new list would be 24 and m+10, respectively. If m were instead replaced by m-8, the medi... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "8133",
"split": "train"
} |
math | [
{
"content": "A projection takes $\\begin{pmatrix} 1 \\\\ -2 \\end{pmatrix}$ to $\\begin{pmatrix} \\frac{3}{2} \\\\ -\\frac{3}{2} \\end{pmatrix}.$ Which vector does the projection take $\\begin{pmatrix} -4 \\\\ 1 \\end{pmatrix}$ to? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} -5/2 \\\\ 5/2 \\end{pmatrix}",
"style": "rule"
} | {
"index": "8134",
"split": "train"
} |
math | [
{
"content": "Find the largest \\( n \\) such that the sum of the fourth powers of any \\( n \\) prime numbers greater than 10 is divisible by \\( n \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "8135",
"split": "train"
} |
math | [
{
"content": "A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the ... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "8136",
"split": "train"
} |
math | [
{
"content": "The interior angles of a convex polygon form an arithmetic sequence, with the smallest angle being $120^\\circ$ and the common difference being $5^\\circ$. Determine the number of sides $n$ for the polygon. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "n = 9",
"style": "rule"
} | {
"index": "8137",
"split": "train"
} |
math | [
{
"content": "One digit of the decimal representation of $\\frac{5}{7}$ is randomly selected. What is the probability that the digit is a 4? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "8138",
"split": "train"
} |
math | [
{
"content": "In each cell of a $15 \\times 15$ table, the number $-1, 0,$ or $+1$ is written such that the sum of the numbers in any row is nonpositive and the sum of the numbers in any column is nonnegative. What is the minimum number of zeros that can be written in the cells of the table? Please output the f... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "8139",
"split": "train"
} |
math | [
{
"content": "When two fair 8-sided dice (labeled from 1 to 8) are tossed, the numbers \\(a\\) and \\(b\\) are obtained. What is the probability that the two-digit number \\(ab\\) (where \\(a\\) and \\(b\\) are digits) and both \\(a\\) and \\(b\\) are divisible by 4? Please output the final answer within \\boxe... | {
"ground_truth": "\\frac{1}{16}",
"style": "rule"
} | {
"index": "8140",
"split": "train"
} |
math | [
{
"content": "Evaluate \\[\\frac{3}{\\log_8{5000^4}} + \\frac{2}{\\log_9{5000^4}},\\] giving your answer as a fraction in lowest terms. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "8141",
"split": "train"
} |
math | [
{
"content": "What is the sum of all the integers from 100 to 2000? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1996050",
"style": "rule"
} | {
"index": "8142",
"split": "train"
} |
math | [
{
"content": "Evaluate: $6 - 5\\left[7 - (\\sqrt{16} + 2)^2\\right] \\cdot 3.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-429",
"style": "rule"
} | {
"index": "8143",
"split": "train"
} |
math | [
{
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-11.5, 11.5]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "552",
"style": "rule"
} | {
"index": "8144",
"split": "train"
} |
math | [
{
"content": "Find the smallest constant $C$ such that for all real numbers $x, y, z$ satisfying $x + y + z = -1$, the following inequality holds:\n$$\n\\left|x^3 + y^3 + z^3 + 1\\right| \\leqslant C \\left|x^5 + y^5 + z^5 + 1\\right|.\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{9}{10}",
"style": "rule"
} | {
"index": "8145",
"split": "train"
} |
math | [
{
"content": "$(100^2-99^2) + (98^2-97^2) + \\ldots + (2^2-1^2) = \\ $ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5050",
"style": "rule"
} | {
"index": "8146",
"split": "train"
} |
math | [
{
"content": "For a positive integer \\( n \\), define \\( s(n) \\) as the smallest positive integer \\( t \\) such that \\( n \\) is a factor of \\( t! \\). Compute the number of positive integers \\( n \\) for which \\( s(n) = 13 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "792",
"style": "rule"
} | {
"index": "8147",
"split": "train"
} |
math | [
{
"content": "How many $3$-digit positive integers have digits whose product equals $30$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "8148",
"split": "train"
} |
math | [
{
"content": "Among all pairs of real numbers $(x, y)$ such that $\\cos \\sin x = \\cos \\sin y$ with $-\\frac{15\\pi}{2} \\le x, y \\le \\frac{15\\pi}{2}$, Ana randomly selects a pair $(X, Y)$. Compute the probability that $X = Y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "8149",
"split": "train"
} |
math | [
{
"content": "In the numbers from $1$ to $2002$, the number of positive integers that contain exactly one digit $0$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "414",
"style": "rule"
} | {
"index": "8150",
"split": "train"
} |
math | [
{
"content": "Let $S(x)$ denote the sum of the digits of a positive integer $x$. Find the maximum possible value of $S(x+2019)-S(x)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "8151",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle XYZ$ be a right triangle with $Y$ as the right angle. A circle with diameter $YZ$ intersects side $XZ$ at $W$. If $XW = 3$ and $YW = 9$, find the length of $WZ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "8152",
"split": "train"
} |
math | [
{
"content": "When the repeating decimal $0.\\overline{12}$ is expressed as a common fraction in lowest terms, what is the sum of its numerator and denominator? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "8153",
"split": "train"
} |
math | [
{
"content": "In a zoo, oranges, bananas, and coconuts were brought to feed three monkeys, with an equal number of each type of fruit. The first monkey was fed only oranges and bananas, with the number of bananas being 40% more than the number of oranges. The second monkey was fed only bananas and coconuts, wit... | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "8154",
"split": "train"
} |
math | [
{
"content": "The Dyslexian alphabet consists of consonants and vowels. It so happens that a finite sequence of letters is a word in Dyslexian precisely if it alternates between consonants and vowels (it may begin with either). There are 4800 five-letter words in Dyslexian. How many letters are in the alphabet?... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "8155",
"split": "train"
} |
math | [
{
"content": "A rectangular prism has dimensions 8 inches by 2 inches by 32 inches. If a cube has the same volume as the prism, what is the surface area of the cube, in square inches? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "384",
"style": "rule"
} | {
"index": "8156",
"split": "train"
} |
math | [
{
"content": "Given that there are 5 advertisements (3 commercial advertisements and 2 Olympic promotional advertisements), the last advertisement is an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively, determine the number of different broadcas... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "8157",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), \\(\\angle A = 60^\\circ\\). Points \\(M\\) and \\(N\\) are on sides \\(AB\\) and \\(AC\\) respectively, such that the circumcenter of triangle \\(ABC\\) bisects segment \\(MN\\). Find the ratio \\(AN:MB\\). Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "2:1",
"style": "rule"
} | {
"index": "8158",
"split": "train"
} |
math | [
{
"content": "Which number is closest to \\(-3.4\\) on a number line? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "8159",
"split": "train"
} |
math | [
{
"content": "Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one \"wall\" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of t... | {
"ground_truth": "$(6,2,1)$",
"style": "rule"
} | {
"index": "8160",
"split": "train"
} |
math | [
{
"content": "Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4$ cents per glob and $J$ blobs of jam at $5$ cents per blob. The cost of the peanut butter and jam to make all the sandwiches is $\\$2.53$. Assume that $B$, $J$, and $N$ are positive integers wi... | {
"ground_truth": "\\$1.65",
"style": "rule"
} | {
"index": "8161",
"split": "train"
} |
math | [
{
"content": "Let positive numbers $x$ and $y$ satisfy: $x > y$, $x+2y=3$. Find the minimum value of $\\frac{1}{x-y} + \\frac{9}{x+5y}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{3}",
"style": "rule"
} | {
"index": "8162",
"split": "train"
} |
math | [
{
"content": "On each cell of a $200 \\times 200$ grid, we place a car, which faces in one of the four cardinal directions. In a move, one chooses a car that does not have a car immediately in front of it, and slides it one cell forward. If a move would cause a car to exit the grid, the car is removed instead. ... | {
"ground_truth": "6014950",
"style": "rule"
} | {
"index": "8163",
"split": "train"
} |
math | [
{
"content": "Tom, Sara, and Jim split $\\$1200$ among themselves to be used in investments. Each starts with a different amount. At the end of one year, they have a total of $\\$1800$. Sara triples her money by the end of the year, Jim doubles his money, whereas Tom loses $\\$200$. What was Tom's original amou... | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "8164",
"split": "train"
} |
math | [
{
"content": "We can view these conditions as a geometry diagram as seen below. So, we know that $\\frac{e}{f}=\\frac{3}{4}$ (since $e=a-b=\\frac{3}{4} c-\\frac{3}{4} d=\\frac{3}{4} f$ and we know that $\\sqrt{e^{2}+f^{2}}=15$ (since this is $\\left.\\sqrt{a^{2}+c^{2}}-\\sqrt{b^{2}+d^{2}}\\right)$. Also, note t... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "8165",
"split": "train"
} |
math | [
{
"content": "Given points $F_{1}$, $F_{2}$ are the foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), and point $M(x_{0},y_{0})$ ($x_{0} > 0$, $y_{0} > 0$) lies on the asymptote of the hyperbola such that $MF_{1} \\perp MF_{2}$. If there is a parabola with focus $F_{2}$ describ... | {
"ground_truth": "2 + \\sqrt{5}",
"style": "rule"
} | {
"index": "8166",
"split": "train"
} |
math | [
{
"content": "The Hangzhou Asian Games are underway, and table tennis, known as China's \"national sport,\" is receiving a lot of attention. In table tennis matches, each game is played to 11 points, with one point awarded for each winning shot. In a game, one side serves two balls first, followed by the other ... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "8167",
"split": "train"
} |
math | [
{
"content": "What is the largest difference that can be formed by subtracting two numbers chosen from the set $\\{ -16,-4,0,2,4,12 \\}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "8168",
"split": "train"
} |
math | [
{
"content": "Define $n!!$ as in the original problem. Evaluate $\\sum_{i=1}^{5} \\frac{(2i-1)!!}{(2i)!!}$, and express the result as a fraction in lowest terms. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{437}{256}",
"style": "rule"
} | {
"index": "8169",
"split": "train"
} |
math | [
{
"content": "Given four non-coplanar points \\(A, B, C, D\\) in space where the distances between any two points are distinct, consider a plane \\(\\alpha\\) that satisfies the following properties: The distances from three of the points \\(A, B, C, D\\) to \\(\\alpha\\) are equal, while the distance from the ... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "8170",
"split": "train"
} |
math | [
{
"content": "Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=101$. What is $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "8171",
"split": "train"
} |
math | [
{
"content": "Calculate the area of an isosceles triangle if the length of its altitude, drawn to one of the equal sides, is $12 \\mathrm{~cm}$, and the length of the base is $15 \\mathrm{~cm}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "8172",
"split": "train"
} |
math | [
{
"content": "Given the set $M = \\{1,2,3,4\\}$, let $A$ be a subset of $M$. The product of all elements in set $A$ is called the \"cumulative value\" of set $A$. It is stipulated that if set $A$ has only one element, its cumulative value is the value of that element, and the cumulative value of the empty set i... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "8173",
"split": "train"
} |
math | [
{
"content": "In the diagram, \\(PQRS\\) is a square with side length 8. Points \\(T\\) and \\(U\\) are on \\(PS\\) and \\(QR\\) respectively with \\(QU = TS = 1\\). The length of \\(TU\\) is closest to Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "8174",
"split": "train"
} |
math | [
{
"content": "Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $16^n$ inclusive, considering a hexadecimal system in which digits range from 1 to 15. Find the integer $n$ for which $T_n$ becomes an integer. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "15015",
"style": "rule"
} | {
"index": "8175",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate plane \\( xOy \\), the coordinates of point \\( F \\) are \\((1,0)\\), and points \\( A \\) and \\( B \\) lie on the parabola \\( y^2 = 4x \\). It is given that \\( \\overrightarrow{OA} \\cdot \\overrightarrow{OB} = -4 \\) and \\( |\\overrightarrow{FA}| - |\\overrightar... | {
"ground_truth": "-11",
"style": "rule"
} | {
"index": "8176",
"split": "train"
} |
math | [
{
"content": "The proportion of top-quality products at this enterprise is $31 \\%$. What is the most likely number of top-quality products in a randomly selected batch of 75 products? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "8177",
"split": "train"
} |
math | [
{
"content": "Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2028",
"style": "rule"
} | {
"index": "8178",
"split": "train"
} |
math | [
{
"content": "Real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x$, $y$, and $z$ are within 1 unit of each other is greater than $\\frac {1}{2}$. What is the smallest possible value of $n$? Please output... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "8179",
"split": "train"
} |
math | [
{
"content": "Given that $\\overrightarrow{a}=(2,3)$, $\\overrightarrow{b}=(-4,7)$, and $\\overrightarrow{a}+\\overrightarrow{c}=\\overrightarrow{0}$, find the projection of $\\overrightarrow{c}$ on the direction of $\\overrightarrow{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "-\\frac{\\sqrt{65}}{5}",
"style": "rule"
} | {
"index": "8180",
"split": "train"
} |
math | [
{
"content": "My school's Physics Club has 22 members. It needs to select 3 officers: chairman, vice-chairman, and sergeant-at-arms. Each person can hold at most one office. Two of the members, Penelope and Quentin, will only be officers if the other one is also an officer. (In other words, either both Pene... | {
"ground_truth": "6960",
"style": "rule"
} | {
"index": "8181",
"split": "train"
} |
math | [
{
"content": "A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $ \\frac{r-\\sqrt{s}}{t}$, where $r$, $s$, a... | {
"ground_truth": "330",
"style": "rule"
} | {
"index": "8182",
"split": "train"
} |
math | [
{
"content": "500 × 3986 × 0.3986 × 5 = ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.25 \\times 3986^2",
"style": "rule"
} | {
"index": "8183",
"split": "train"
} |
math | [
{
"content": "Each triangle in a sequence is either a 30-60-90 triangle or a 45-45-90 triangle. The hypotenuse of each 30-60-90 triangle serves as the longer leg of the adjacent 30-60-90 triangle, except for the final triangle which is a 45-45-90 triangle. The hypotenuse of the largest triangle is 16 centimeter... | {
"ground_truth": "\\frac{6\\sqrt{6}}{2}",
"style": "rule"
} | {
"index": "8184",
"split": "train"
} |
math | [
{
"content": "Given a cone whose vertex and the circumference of its base are both on the same sphere, if the radius of the sphere is $1$, then when the volume of the cone is maximized, the height of the cone is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{4}{3}",
"style": "rule"
} | {
"index": "8185",
"split": "train"
} |
math | [
{
"content": "Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1+3\\sqrt{2}",
"style": "rule"
} | {
"index": "8186",
"split": "train"
} |
math | [
{
"content": "Given that $\\cos \\alpha =\\dfrac{4}{5}$ and $\\cos (\\alpha +\\beta )=\\dfrac{5}{13}$, where $\\alpha$ and $\\beta$ are acute angles.\n1. Find the value of $\\sin 2\\alpha$.\n2. Find the value of $\\sin \\beta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{33}{65}",
"style": "rule"
} | {
"index": "8187",
"split": "train"
} |
math | [
{
"content": "The expression $\\frac{4k+8}{4}$ simplifies to an expression of the form $ak+b$ where $a$ and $b$ are integers. Find $\\frac{a}{b}$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "8188",
"split": "train"
} |
math | [
{
"content": "Find the smallest possible value of the expression $$\\frac{(a+b)^2+(b-c)^2+(c-a)^2}{b^2},$$where $b > c > a$ are real numbers, and $b \\neq 0.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "8189",
"split": "train"
} |
math | [
{
"content": "Each side of a cube has a stripe drawn diagonally from one vertex to the opposite vertex. The stripes can either go from the top-right to bottom-left or from top-left to bottom-right, chosen at random for each face. What is the probability that there is at least one continuous path following the s... | {
"ground_truth": "\\frac{3}{64}",
"style": "rule"
} | {
"index": "8190",
"split": "train"
} |
math | [
{
"content": "Determine the number of solutions to the equation\n\\[\\tan (7 \\pi \\cos \\theta) = \\cot (3 \\pi \\sin \\theta)\\] where $\\theta \\in (0, 3\\pi).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "8191",
"split": "train"
} |
math | [
{
"content": "Is there a number in which every digit from 1 to 9 is used exactly once, and for which the number formed by the first $k$ digits is divisible by $k$ for every $k$ from 1 to 9? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "381654729",
"style": "rule"
} | {
"index": "8192",
"split": "train"
} |
math | [
{
"content": "Given that $\\{1, a, \\frac{b}{a}\\} = \\{0, a^2, a+b\\}$, find the value of $a^{2015} + b^{2014}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "8193",
"split": "train"
} |
math | [
{
"content": "Teacher Zhang led the students of class 6 (1) to plant trees. The students can be divided into 5 equal groups. It is known that each teacher and student plants the same number of trees, with a total of 527 trees planted. How many students are there in class 6 (1)? Please output the final answer wi... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "8194",
"split": "train"
} |
math | [
{
"content": "Restore the digits. On the board, the product of three consecutive even numbers was written. During the break, Vasya erased some of the digits. As a result, the board shows $87*****8$. Help Petya find the missing digits in the product. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "87526608",
"style": "rule"
} | {
"index": "8195",
"split": "train"
} |
math | [
{
"content": "A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024$. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the firs... | {
"ground_truth": "342",
"style": "rule"
} | {
"index": "8196",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\lceil-2.4\\rceil$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "8197",
"split": "train"
} |
math | [
{
"content": "How many positive three-digit integers less than 500 have at least two digits that are the same? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "8198",
"split": "train"
} |
math | [
{
"content": "A and B play a game as follows. Each throws a dice. Suppose A gets \\(x\\) and B gets \\(y\\). If \\(x\\) and \\(y\\) have the same parity, then A wins. If not, they make a list of all two-digit numbers \\(ab \\leq xy\\) with \\(1 \\leq a, b \\leq 6\\). Then they take turns (starting with A) repla... | {
"ground_truth": "3/4",
"style": "rule"
} | {
"index": "8199",
"split": "train"
} |
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