data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Find the number of all trees planted at a five-foot distance from each other on a rectangular plot of land, the sides of which are 120 feet and 70 feet. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "375",
"style": "rule"
} | {
"index": "7300",
"split": "train"
} |
math | [
{
"content": "For the function $f(x)= \\sqrt {2}(\\sin x+\\cos x)$, the following four propositions are given: \n$(1)$ There exists $\\alpha\\in\\left(- \\frac {\\pi}{2},0\\right)$, such that $f(\\alpha)= \\sqrt {2}$; \n$(2)$ The graph of the function $f(x)$ is symmetric about the line $x=- \\frac {3\\pi}{4}$... | {
"ground_truth": "(2)(3)",
"style": "rule"
} | {
"index": "7301",
"split": "train"
} |
math | [
{
"content": "A three-digit number \\( X \\) was composed of three different digits, \\( A, B, \\) and \\( C \\). Four students made the following statements:\n\n- Petya: \"The largest digit in the number \\( X \\) is \\( B \\).\"\n- Vasya: \"\\( C = 8 \\).\"\n- Tolya: \"The largest digit is \\( C \\).\"\n- Dim... | {
"ground_truth": "798",
"style": "rule"
} | {
"index": "7302",
"split": "train"
} |
math | [
{
"content": "A tree doubled its height every year until it reached a height of 32 feet at the end of 6 years. What was the height of the tree, in feet, at the end of 3 years? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4 \\text{ feet}",
"style": "rule"
} | {
"index": "7303",
"split": "train"
} |
math | [
{
"content": "Two sides of an isosceles triangle are 10 inches and 20 inches. If the shortest side of a similar triangle is 50 inches, what is the perimeter of the larger triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "250\\text{ inches}",
"style": "rule"
} | {
"index": "7304",
"split": "train"
} |
math | [
{
"content": "Three fair, standard six-sided dice are rolled. What is the probability that the sum of the numbers on the top faces is 18? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{216}",
"style": "rule"
} | {
"index": "7305",
"split": "train"
} |
math | [
{
"content": "A positive integer is called primer if it has a prime number of distinct prime factors. A positive integer is called primest if it has a primer number of distinct primer factors. A positive integer is called prime-minister if it has a primest number of distinct primest factors. Let $N$ be the smal... | {
"ground_truth": "378000",
"style": "rule"
} | {
"index": "7306",
"split": "train"
} |
math | [
{
"content": "How many four-digit integers $abcd$, with $a \\neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "7307",
"split": "train"
} |
math | [
{
"content": "According to the classification standard of the Air Pollution Index (API) for city air quality, when the air pollution index is not greater than 100, the air quality is good. The environmental monitoring department of a city randomly selected the air pollution index for 5 days from last month's ai... | {
"ground_truth": "440",
"style": "rule"
} | {
"index": "7308",
"split": "train"
} |
math | [
{
"content": "The number \\( x \\) is such that \\( \\log _{2}\\left(\\log _{4} x\\right) + \\log _{4}\\left(\\log _{8} x\\right) + \\log _{8}\\left(\\log _{2} x\\right) = 1 \\). Find the value of the expression \\( \\log _{4}\\left(\\log _{2} x\\right) + \\log _{8}\\left(\\log _{4} x\\right) + \\log _{2}\\left... | {
"ground_truth": "0.87",
"style": "rule"
} | {
"index": "7309",
"split": "train"
} |
math | [
{
"content": "In triangle $AHI$, which is equilateral, lines $\\overline{BC}$, $\\overline{DE}$, and $\\overline{FG}$ are all parallel to $\\overline{HI}$. The lengths satisfy $AB = BD = DF = FH$. However, the point $F$ on line segment $AH$ is such that $AF$ is half of $AH$. Determine the ratio of the area of t... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "7310",
"split": "train"
} |
math | [
{
"content": "A laptop is originally priced at $\\$1200$. It is on sale for $15\\%$ off. John applies two additional coupons: one gives $10\\%$ off the discounted price, and another gives $5\\%$ off the subsequent price. What single percent discount would give the same final price as these three successive disc... | {
"ground_truth": "27.325\\%",
"style": "rule"
} | {
"index": "7311",
"split": "train"
} |
math | [
{
"content": "In an olympiad, 2006 students participated. It was found that a student, Vasia, solved only one out of the six problems. Additionally, the number of participants who solved at least 1 problem is 4 times greater than those who solved at least 2 problems;\nthe number who solved at least 2 problems i... | {
"ground_truth": "982",
"style": "rule"
} | {
"index": "7312",
"split": "train"
} |
math | [
{
"content": "Points $A, B, C, D$ lie on a circle in that order such that $\\frac{A B}{B C}=\\frac{D A}{C D}$. If $A C=3$ and $B D=B C=4$, find $A D$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "7313",
"split": "train"
} |
math | [
{
"content": "Given that $a,b$ are positive real numbers, and $({(a-b)}^{2}=4{{(ab)}^{3}})$, find the minimum value of $\\dfrac{1}{a}+\\dfrac{1}{b}$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "7314",
"split": "train"
} |
math | [
{
"content": "What is the ratio of the volume of cone $C$ to the volume of cone $D$? Express your answer as a common fraction. Cone $C$ has a radius of 16.4 and height of 30.5, while cone $D$ has a radius of 30.5 and height of 16.4. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{164}{305}",
"style": "rule"
} | {
"index": "7315",
"split": "train"
} |
math | [
{
"content": "There are 111 balls in a box, each being red, green, blue, or white. It is known that if 100 balls are drawn, it ensures getting balls of all four colors. Find the smallest integer $N$ such that if $N$ balls are drawn, it can ensure getting balls of at least three different colors. Please output t... | {
"ground_truth": "88",
"style": "rule"
} | {
"index": "7316",
"split": "train"
} |
math | [
{
"content": "Eddy draws $6$ cards from a standard $52$ -card deck. What is the probability that four of the cards that he draws have the same value? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3/4165",
"style": "rule"
} | {
"index": "7317",
"split": "train"
} |
math | [
{
"content": "Given $a, b \\in \\mathbb{R}$, $m = ab + 1$, $n = a + b$.\n- $(1)$ If $a > 1$, $b < 1$, compare the sizes of $m$ and $n$.\n- $(2)$ If $a > 1$, $b > 1$, and $m - n = 49$, find the minimum value of $a + b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "7318",
"split": "train"
} |
math | [
{
"content": "Given an odd function $f(x)$ on $\\mathbb{R}$, for any $x \\in \\mathbb{R}$, $f(x+1) = -f(x)$, and when $x \\in (-1, 1)$, $f(x) = x$. Find the value of $f(3) + f(-7.5)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "7319",
"split": "train"
} |
math | [
{
"content": "In triangle $VWX$ shown, $VX = \\sqrt{13}$ and $VW = 3$. What is $\\tan V$?\n\n[asy]\npair V,W,X;\nW = (0,0);\nV = (0,3);\nX = (2,0);\ndraw(V--W--X--V);\ndraw(rightanglemark(V,W,X,7));\nlabel(\"$W$\",W,SW);\nlabel(\"$V$\",V,NW);\nlabel(\"$X$\",X,SE);\nlabel(\"$3$\",(V+W)/2,NW);\nlabel(\"$\\sqrt{13... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "7320",
"split": "train"
} |
math | [
{
"content": "Jessica has three marbles colored red, green, and blue. She randomly selects a non-empty subset of them (such that each subset is equally likely) and puts them in a bag. You then draw three marbles from the bag with replacement. The colors you see are red, blue, red. What is the probability that t... | {
"ground_truth": "\\frac{27}{35}",
"style": "rule"
} | {
"index": "7321",
"split": "train"
} |
math | [
{
"content": "Consider four functions, labelled from (6) to (9). The domain of function (7) is $\\{-6, -5, -4, -3, -2, -1, 0, 1\\}$. Determine the product of the labels of the functions that are invertible. The graphs are as follows: \n- Function (6): $y = x^3 - 3x$\n- Function (7): A set of discrete points: $\... | {
"ground_truth": "504",
"style": "rule"
} | {
"index": "7322",
"split": "train"
} |
math | [
{
"content": "Given the polynomial with integer coefficients:\n\\[ f(x) = x^5 + a_1 x^4 + a_2 x^3 + a_3 x^2 + a_4 x + a_5 \\]\n\nIf \\( f(\\sqrt{3} + \\sqrt{2}) = 0 \\) and \\( f(1) + f(3) = 0 \\), find \\( f(-1) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "7323",
"split": "train"
} |
math | [
{
"content": "Given that\n\\begin{eqnarray*}&(1)& x\\text{ and }y\\text{ are both integers between 100 and 999, inclusive;}\\qquad \\qquad \\qquad \\qquad \\qquad \\\\ &(2)& y\\text{ is the number formed by reversing the digits of }x\\text{; and}\\\\ &(3)& z=|x-y|. \\end{eqnarray*}\n How many distinct values of... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "7324",
"split": "train"
} |
math | [
{
"content": "The Gnollish language consists of 3 words, ``splargh,'' ``glumph,'' and ``amr.'' In a sentence, ``splargh'' cannot come directly before ``glumph''; all other sentences are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Gnollish? Plea... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "7325",
"split": "train"
} |
math | [
{
"content": "In the plane Cartesian coordinate system \\( xOy \\), a moving line \\( l \\) is tangent to the parabola \\( \\Gamma: y^{2} = 4x \\), and intersects the hyperbola \\( \\Omega: x^{2} - y^{2} = 1 \\) at one point on each of its branches, left and right, labeled \\( A \\) and \\( B \\). Find the mini... | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": "7326",
"split": "train"
} |
math | [
{
"content": "If a sequence of numbers \\(a_{1}, a_{2}, \\cdots\\) satisfies, for any positive integer \\(n\\),\n$$\na_{n}=\\frac{n^{2}+n-2-\\sqrt{2}}{n^{2}-2},\n$$\n\nthen what is the value of \\(a_{1} a_{2} \\cdots a_{2016}\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2016\\sqrt{2} - 2015",
"style": "rule"
} | {
"index": "7327",
"split": "train"
} |
math | [
{
"content": "Let $r$, $s$, and $t$ be the three roots of the equation $$\n8x^3 + 1001x + 2008 = 0.\n$$Find $(r + s)^3 + (s + t)^3 + (t + r)^3.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "753",
"style": "rule"
} | {
"index": "7328",
"split": "train"
} |
math | [
{
"content": "A regular octagon's perimeter is given as $P=16\\sqrt{2}$. If $R_i$ denotes the midpoint of side $V_iV_{i+1}$ (with $V_8V_1$ as the last side), calculate the area of the quadrilateral $R_1R_3R_5R_7$.\nA) $8$\nB) $4\\sqrt{2}$\nC) $10$\nD) $8 + 4\\sqrt{2}$\nE) $12$ Please output the final answer wit... | {
"ground_truth": "8 + 4\\sqrt{2}",
"style": "rule"
} | {
"index": "7329",
"split": "train"
} |
math | [
{
"content": "Let the function \\( f(x) = \\left| \\log_{2} x \\right| \\). Real numbers \\( a \\) and \\( b \\) (where \\( a < b \\)) satisfy the following conditions:\n\\[\n\\begin{array}{l}\nf(a+1) = f(b+2), \\\\\nf(10a + 6b + 22) = 4\n\\end{array}\n\\]\n\nFind \\( ab \\). Please output the final answer with... | {
"ground_truth": "2/15",
"style": "rule"
} | {
"index": "7330",
"split": "train"
} |
math | [
{
"content": "It is known that $x^5 = a_0 + a_1 (1+x) + a_2 (1+x)^2 + a_3 (1+x)^3 + a_4 (1+x)^4 + a_5 (1+x)^5$, find the value of $a_0 + a_2 + a_4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-16",
"style": "rule"
} | {
"index": "7331",
"split": "train"
} |
math | [
{
"content": "Let $P$ and $Q$ be points on line $l$ with $P Q=12$. Two circles, $\\omega$ and $\\Omega$, are both tangent to $l$ at $P$ and are externally tangent to each other. A line through $Q$ intersects $\\omega$ at $A$ and $B$, with $A$ closer to $Q$ than $B$, such that $A B=10$. Similarly, another line t... | {
"ground_truth": "\\frac{8}{9}",
"style": "rule"
} | {
"index": "7332",
"split": "train"
} |
math | [
{
"content": "What is the greatest possible sum of the digits in the base-eight representation of a positive integer less than $1728$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "7333",
"split": "train"
} |
math | [
{
"content": "The roots of\n\\[z^7 = -\\frac{1}{\\sqrt{2}} - \\frac{i}{\\sqrt{2}}\\]are $\\text{cis } \\theta_1$, $\\text{cis } \\theta_2$, $\\dots$, $\\text{cis } \\theta_7$, where $0^\\circ \\le \\theta_k < 360^\\circ$ for all $1 \\le k \\le 7$. Find $\\theta_1 + \\theta_2 + \\dots + \\theta_7$. Give your a... | {
"ground_truth": "1305^\\circ",
"style": "rule"
} | {
"index": "7334",
"split": "train"
} |
math | [
{
"content": "An element is randomly chosen from among the first $15$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is $1$?\n\nNote: The 1 at the top is often labelled the \"zeroth\" row of Pascal's Triangle, by convention. So to count a total of 15 rows, use rows 0 th... | {
"ground_truth": "\\frac{29}{120}",
"style": "rule"
} | {
"index": "7335",
"split": "train"
} |
math | [
{
"content": "Sequentially throw a die and record the numbers obtained as $a$ and $b$. Perform the following operations: \n```\nINPUT \"a, b=\"; a, b \nIF a >= b THEN \ny = a - b \nELSE \ny = b - a \nEND IF \nPRINT y \nEND \n```\n(1) If $a=3$, $b=6$, find the value of $y$ output by the computer after r... | {
"ground_truth": "\\frac{2}{9}",
"style": "rule"
} | {
"index": "7336",
"split": "train"
} |
math | [
{
"content": "Compute the number of ways to select 99 cells of a $19 \\times 19$ square grid such that no two selected cells share an edge or vertex. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": "7337",
"split": "train"
} |
math | [
{
"content": "Determine the number of different ways to schedule volleyball, basketball, and table tennis competitions in 4 different gyms, given that each sport must be held in only one gym and that no more than 2 sports can take place in the same gym. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "7338",
"split": "train"
} |
math | [
{
"content": "In the equation, $\\overline{\\mathrm{ABCD}}+\\overline{\\mathrm{EFG}}=2020$, different letters represent different digits. What is $A+B+C+D+E+F+G=$ $\\qquad$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "7339",
"split": "train"
} |
math | [
{
"content": "Given the set $A=\\{x|1 < x < k\\}$ and the set $B=\\{y|y=2x-5, x \\in A\\}$, if $A \\cap B = \\{x|1 < x < 2\\}$, find the value of the real number $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3.5",
"style": "rule"
} | {
"index": "7340",
"split": "train"
} |
math | [
{
"content": "If $7=x^2+\\frac{1}{x^2}$, then what is the greatest possible value of $x+\\frac{1}{x}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "7341",
"split": "train"
} |
math | [
{
"content": "(The full score for this question is 12 points) Given $f(x+1)=x^2-1$, \n(1) Find $f(x)$ \n(2) Find the maximum and minimum values of $f(x)$, and specify the corresponding values of $x$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "7342",
"split": "train"
} |
math | [
{
"content": "The expression $x^2 - 16x + 60$ can be written in the form $(x - a)(x - b)$, where $a$ and $b$ are both nonnegative integers and $a > b$. What is the value of $3b - a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "7343",
"split": "train"
} |
math | [
{
"content": "A quadrilateral in the plane has vertices $(1, 3)$, $(1, 1)$, $(2, 1)$ and $(2006, 2007)$. How many square units is the area of the quadrilateral? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3008",
"style": "rule"
} | {
"index": "7344",
"split": "train"
} |
math | [
{
"content": "Let $a,b,c$ be positive real numbers such that $a+b+c=10$ and $ab+bc+ca=25$. Let $m=\\min\\{ab,bc,ca\\}$. Find the largest possible value of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{25}{9}",
"style": "rule"
} | {
"index": "7345",
"split": "train"
} |
math | [
{
"content": "Given an equilateral triangle \\( \\triangle ABC \\) with a side length of 1,\n\\[\n\\overrightarrow{AP} = \\frac{1}{3}(\\overrightarrow{AB} + \\overrightarrow{AC}), \\quad \\overrightarrow{AQ} = \\overrightarrow{AP} + \\frac{1}{2}\\overrightarrow{BC}.\n\\]\n\nFind the area of \\( \\triangle APQ \... | {
"ground_truth": "\\frac{\\sqrt{3}}{12}",
"style": "rule"
} | {
"index": "7346",
"split": "train"
} |
math | [
{
"content": "A bus arrives randomly sometime between 2:00 and 3:00 and waits for 15 minutes before leaving. If Carla arrives randomly between 2:00 and 3:00, what is the probability that the bus will still be there when Carla arrives? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{32}",
"style": "rule"
} | {
"index": "7347",
"split": "train"
} |
math | [
{
"content": "Fifteen distinct points are designated on $\\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\\overline{AB}$; $4$ other points on side $\\overline{BC}$; and $5$ other points on side $\\overline{CA}$. Find the number of triangles with positive area whose vertices are amon... | {
"ground_truth": "390",
"style": "rule"
} | {
"index": "7348",
"split": "train"
} |
math | [
{
"content": "Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} | {
"index": "7349",
"split": "train"
} |
math | [
{
"content": "Put ping pong balls in 10 boxes. The number of balls in each box must not be less than 11, must not be 17, must not be a multiple of 6, and must be different from each other. What is the minimum number of ping pong balls needed? Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "174",
"style": "rule"
} | {
"index": "7350",
"split": "train"
} |
math | [
{
"content": "How many ways are there to arrange the $6$ permutations of the tuple $(1, 2, 3)$ in a sequence, such that each pair of adjacent permutations contains at least one entry in common? \nFor example, a valid such sequence is given by $(3, 2, 1) - (2, 3, 1) - (2, 1, 3) - (1, 2, 3) - (1, 3, 2) - (3,... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "7351",
"split": "train"
} |
math | [
{
"content": "Determine the sum of the fifth and sixth elements in Row 20 of Pascal's triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20349",
"style": "rule"
} | {
"index": "7352",
"split": "train"
} |
math | [
{
"content": "If $(x^2+1)(2x+1)^9 = a_0 + a_1(x+2) + a_2(x+2)^2 + \\ldots + a_{11}(x+2)^{11}$, then the value of $a_0 + a_1 + \\ldots + a_{11}$ is. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "7353",
"split": "train"
} |
math | [
{
"content": "A ball with a diameter of 6 inches rolls along a complex track from start point A to endpoint B. The track comprises four semicircular arcs with radii $R_1 = 120$ inches, $R_2 = 50$ inches, $R_3 = 90$ inches, and $R_4 = 70$ inches respectively. The ball always stays in contact with the track and r... | {
"ground_truth": "330\\pi",
"style": "rule"
} | {
"index": "7354",
"split": "train"
} |
math | [
{
"content": "Given points \\( A(4,0) \\) and \\( B(2,2) \\) are inside the ellipse \\( \\frac{x^{2}}{25}+\\frac{y^{2}}{9}=1 \\), and \\( M \\) is a point on the ellipse, find the maximum value of \\( |MA| + |MB| \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10 + 2\\sqrt{10}",
"style": "rule"
} | {
"index": "7355",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than 60 have an even number of positive divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "7356",
"split": "train"
} |
math | [
{
"content": "A natural number \\( 1 \\leq n \\leq 221 \\) is called lucky if, when dividing 221 by \\( n \\), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "115",
"style": "rule"
} | {
"index": "7357",
"split": "train"
} |
math | [
{
"content": "The circumcircle of acute $\\triangle ABC$ has center $O$ . The line passing through point $O$ perpendicular to $\\overline{OB}$ intersects lines $AB$ and $BC$ at $P$ and $Q$ , respectively. Also $AB=5$ , $BC=4$ , $BQ=4.5$ , and $BP=\\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive i... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "7358",
"split": "train"
} |
math | [
{
"content": "Given a parabola $y^2 = 2px$ ($p > 0$) and a hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) share a common focus $F$, and point $A$ is the intersection point of the two curves. If $AF \\perp x$-axis, find the eccentricity of the hyperbola. Please output the final answer withi... | {
"ground_truth": "\\sqrt{2} + 1",
"style": "rule"
} | {
"index": "7359",
"split": "train"
} |
math | [
{
"content": "Given that the base edge length of a regular square pyramid is $2$, and the side edge length is $\\sqrt{6}$, determine the volume of the pyramid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{3}",
"style": "rule"
} | {
"index": "7360",
"split": "train"
} |
math | [
{
"content": "The probability that Class A will be assigned exactly 2 of the 8 awards, with each of the 4 classes (A, B, C, and D) receiving at least 1 award is $\\qquad$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{7}",
"style": "rule"
} | {
"index": "7361",
"split": "train"
} |
math | [
{
"content": "How many orbitals contain one or more electrons in an isolated ground state iron atom (Z = 26)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "7362",
"split": "train"
} |
math | [
{
"content": "If $x=3$, $y=2x$, and $z=3y$, what is the average of $x$, $y$, and $z$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "7363",
"split": "train"
} |
math | [
{
"content": "If $0 < \\alpha < \\frac{\\pi}{2}$ and $\\tan \\alpha = 2$, then $\\frac{\\sin 2\\alpha + 1}{\\cos^4 \\alpha - \\sin^4 \\alpha} = \\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "7364",
"split": "train"
} |
math | [
{
"content": "Cylinder $C$'s height is equal to the diameter of cylinder $D$ and cylinder $C$'s diameter is equal to the height $h$ of cylinder $D$. If the volume of cylinder $D$ is three times the volume of cylinder $C$, the volume of cylinder $D$ can be written as $M \\pi h^3$ cubic units. Find the value of $... | {
"ground_truth": "\\frac{9}{4}",
"style": "rule"
} | {
"index": "7365",
"split": "train"
} |
math | [
{
"content": "All the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are written in a 3x3 array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to 18. What is the number in the center? Please output t... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "7366",
"split": "train"
} |
math | [
{
"content": "In ancient China, when determining the pentatonic scale of Gong, Shang, Jiao, Zhi, and Yu, a method of calculation involving a three-part loss and gain was initially used. The second note was obtained by subtracting one-third from the first note, and the third note was obtained by adding one-third... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "7367",
"split": "train"
} |
math | [
{
"content": "In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\\left\\{{\\begin{array}{l}{x=2+3\\cos\\alpha}\\\\{y=3\\sin\\alpha}\\end{array}}\\right.$ ($\\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x... | {
"ground_truth": "\\frac{{3\\sqrt{5}}}{5}",
"style": "rule"
} | {
"index": "7368",
"split": "train"
} |
math | [
{
"content": "Ann and Anne are in bumper cars starting 50 meters apart. Each one approaches the other at a constant ground speed of $10 \\mathrm{~km} / \\mathrm{hr}$. A fly starts at Ann, flies to Anne, then back to Ann, and so on, back and forth until it gets crushed when the two bumper cars collide. When goin... | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "7369",
"split": "train"
} |
math | [
{
"content": "Four princesses each guessed a two-digit number, and Ivan guessed a four-digit number. After they wrote their numbers in a row in some order, they got the sequence 132040530321. Find Ivan's number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5303",
"style": "rule"
} | {
"index": "7370",
"split": "train"
} |
math | [
{
"content": "I have a bag containing red and green marbles. Initially, the ratio of red to green marbles is 3:2. If I remove 18 red marbles and add 15 green marbles, the new ratio becomes 2:5. How many red marbles were there in the bag initially? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "7371",
"split": "train"
} |
math | [
{
"content": "For a real number $x$, let $[x]$ be $x$ rounded to the nearest integer and $\\langle x\\rangle$ be $x$ rounded to the nearest tenth. Real numbers $a$ and $b$ satisfy $\\langle a\\rangle+[b]=98.6$ and $[a]+\\langle b\\rangle=99.3$. Compute the minimum possible value of $[10(a+b)]$. Please output th... | {
"ground_truth": "988",
"style": "rule"
} | {
"index": "7372",
"split": "train"
} |
math | [
{
"content": "The Tasty Candy Company always puts the same number of pieces of candy into each one-pound bag of candy they sell. Mike bought 4 one-pound bags and gave each person in his class 15 pieces of candy. Mike had 23 pieces of candy left over. Betsy bought 5 one-pound bags and gave 23 pieces of candy to ... | {
"ground_truth": "302",
"style": "rule"
} | {
"index": "7373",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=x^{2}+6 x+7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "7374",
"split": "train"
} |
math | [
{
"content": "Mia and Tom jog on a circular track. Mia jogs counterclockwise and completes a lap every 96 seconds, while Tom jogs clockwise and completes a lap every 75 seconds. They both start from the same point at the same time. If a photographer positioned inside the track takes a snapshot sometime between ... | {
"ground_truth": "\\frac{5}{6}",
"style": "rule"
} | {
"index": "7375",
"split": "train"
} |
math | [
{
"content": "A venture capital firm is considering investing in the development of a new energy product, with estimated investment returns ranging from 100,000 yuan to 10,000,000 yuan. The firm is planning to establish a reward scheme for the research team: the reward amount $y$ (in units of 10,000 yuan) incre... | {
"ground_truth": "328",
"style": "rule"
} | {
"index": "7376",
"split": "train"
} |
math | [
{
"content": "Chords \\(AB\\) and \\(CD\\) of a circle with center \\(O\\) both have a length of 5. The extensions of segments \\(BA\\) and \\(CD\\) beyond points \\(A\\) and \\(D\\) intersect at point \\(P\\), where \\(DP=13\\). The line \\(PO\\) intersects segment \\(AC\\) at point \\(L\\). Find the ratio \\(... | {
"ground_truth": "13/18",
"style": "rule"
} | {
"index": "7377",
"split": "train"
} |
math | [
{
"content": "Figure $OPQR$ is a square. Point $O$ is the origin, and point $Q$ has coordinates (2,2). What are the coordinates for $T$ so that the area of triangle $PQT$ equals the area of square $OPQR$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-2,0)",
"style": "rule"
} | {
"index": "7378",
"split": "train"
} |
math | [
{
"content": "The product of the first and third terms of the geometric sequence $\\{a_n\\}$, given that $a_1$ and $a_4$ are the roots of the equation $x^2-2x-3=0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "7379",
"split": "train"
} |
math | [
{
"content": "Arrange the natural numbers according to the following triangular pattern. What is the sum of the numbers in the 10th row?\n\n```\n 1\n 2 3 4\n 5 6 7 8 9\n 10 11 12 13 14 15 16\n17 18 19 20 21 22 23 24 25\n...\n``` Please output the final answ... | {
"ground_truth": "1729",
"style": "rule"
} | {
"index": "7380",
"split": "train"
} |
math | [
{
"content": "Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has 4 goldfish at the same time that Gretel has 128 goldfish, then in how many months from that time will they have the same number of goldfish? Please output the f... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "7381",
"split": "train"
} |
math | [
{
"content": "Let $F(0) = 0,$ $F(1) = \\frac{3}{2},$ and\n\\[F(n) = \\frac{5}{2} F(n - 1) - F(n - 2)\\]for $n \\ge 2.$ Find\n\\[\\sum_{n = 0}^\\infty \\frac{1}{F(2^n)}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "7382",
"split": "train"
} |
math | [
{
"content": "In the number \\(2016*****02*\\), each of the 5 stars must be replaced by any of the digits \\(0, 2, 4, 5, 7, 9\\) (digits can repeat) so that the resulting 11-digit number is divisible by 15. How many ways can this be done? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "864",
"style": "rule"
} | {
"index": "7383",
"split": "train"
} |
math | [
{
"content": "A flag consists of three horizontal strips of fabric, each of a solid color, from the choices of red, white, blue, green, or yellow. If no two adjacent strips can be the same color, and an additional rule that no color can be used more than twice, how many distinct flags are possible? Please outpu... | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "7384",
"split": "train"
} |
math | [
{
"content": "All dwarves are either liars or knights. Liars always lie, while knights always tell the truth. Each cell of a $4 \\times 4$ board contains one dwarf. It is known that among them there are both liars and knights. Each dwarf stated: \"Among my neighbors (by edge), there are an equal number of liars... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "7385",
"split": "train"
} |
math | [
{
"content": "A right triangle has legs measuring 20 inches and 21 inches. What is the length of the hypotenuse, in inches? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "7386",
"split": "train"
} |
math | [
{
"content": "Which letter will go in the square marked with $*$ in the grid where each of the letters A, B, C, D, and E appears exactly once in each row and column? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "B",
"style": "rule"
} | {
"index": "7387",
"split": "train"
} |
math | [
{
"content": "Two spheres are inscribed in a dihedral angle in such a way that they touch each other. The radius of one sphere is 1.5 times the radius of the other, and the line connecting the centers of the spheres forms an angle of $30^{\\circ}$ with the edge of the dihedral angle. Find the measure of the dih... | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "7388",
"split": "train"
} |
math | [
{
"content": "A firecracker was thrown vertically upwards with a speed of 20 m/s. Three seconds after the start of its flight, it exploded into two unequal parts, the mass ratio of which is $1: 2$. The smaller fragment immediately after the explosion flew horizontally at a speed of $16 \\mathrm{~m}/\\mathrm{s}$... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "7389",
"split": "train"
} |
math | [
{
"content": "Suppose $p(x)$ is a monic cubic polynomial with real coefficients such that $p(3-2i)=0$ and $p(0)=-52$.\n\nDetermine $p(x)$ (in expanded form). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "x^3-10x^2+37x-52",
"style": "rule"
} | {
"index": "7390",
"split": "train"
} |
math | [
{
"content": "Given that α and β are acute angles, and $\\tan \\alpha = \\frac{2}{t}$, $\\tan \\beta = \\frac{t}{15}$. When $10\\tan \\alpha + 3\\tan \\beta$ reaches its minimum value, the value of $\\alpha + \\beta$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "7391",
"split": "train"
} |
math | [
{
"content": "What fraction of the entire wall is painted red if Matilda paints half of her section red and Ellie paints one third of her section red? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{12}",
"style": "rule"
} | {
"index": "7392",
"split": "train"
} |
math | [
{
"content": "When the binary number $100101110010_2$ is divided by 4, what is the remainder (give your answer in base 10)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "7393",
"split": "train"
} |
math | [
{
"content": "Simplify $\\cot 10 + \\tan 5.$\n\nEnter your answer as a trigonometric function evaluated at an integer, such as \"sin 7\". Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\csc 10",
"style": "rule"
} | {
"index": "7394",
"split": "train"
} |
math | [
{
"content": "There are 7 people standing in a row. How many different arrangements are there according to the following requirements?\n\n(1) Among them, A cannot stand at the head, and B cannot stand at the tail;\n(2) Among them, A, B, and C must stand next to each other;\n(3) Among them, A, B, and C must not ... | {
"ground_truth": "840",
"style": "rule"
} | {
"index": "7395",
"split": "train"
} |
math | [
{
"content": "A circle has 2017 distinct points $A_{1}, \\ldots, A_{2017}$ marked on it, and all possible chords connecting pairs of these points are drawn. A line is drawn through the point $A_{1}$, which does not pass through any of the points $A_{2}, \\ldots A_{2017}$. Find the maximum possible number of cho... | {
"ground_truth": "1018080",
"style": "rule"
} | {
"index": "7396",
"split": "train"
} |
math | [
{
"content": "Find the integer $n,$ $0 \\le n \\le 180,$ such that $\\cos n^\\circ = \\cos 758^\\circ.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "7397",
"split": "train"
} |
math | [
{
"content": "The number of distinct even numbers that can be formed using the digits 0, 1, 2, and 3. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "7398",
"split": "train"
} |
math | [
{
"content": "Given that 60% of all students in Ms. Hanson's class answered \"Yes\" to the question \"Do you love science\" at the beginning of the school year, 40% answered \"No\", 80% answered \"Yes\" and 20% answered \"No\" at the end of the school year, calculate the difference between the maximum and the m... | {
"ground_truth": "40\\%",
"style": "rule"
} | {
"index": "7399",
"split": "train"
} |
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