data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Which of the following words has the largest value, given that the first five letters of the alphabet are assigned the values $A=1, B=2, C=3, D=4, E=5$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "BEE",
"style": "rule"
} | {
"index": "6300",
"split": "train"
} |
math | [
{
"content": "How many positive whole numbers have cube roots that are less than $10$? For example, $20$ would count since $\\sqrt[3]{20}<10.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "999",
"style": "rule"
} | {
"index": "6301",
"split": "train"
} |
math | [
{
"content": "Given a right triangle where one leg has length $4x + 2$ feet and the other leg has length $(x-3)^2$ feet, find the value of $x$ if the hypotenuse is $5x + 1$ feet. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{\\frac{3}{2}}",
"style": "rule"
} | {
"index": "6302",
"split": "train"
} |
math | [
{
"content": "Three congruent isosceles triangles $DAO$, $AOB$ and $OBC$ have $AD=AO=OB=BC=10$ and $AB=DO=OC=12$. These triangles are arranged to form trapezoid $ABCD$, as shown. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB$.\n\n[asy]\npair A, B, C, D, O, P;\nA= (6, 8);\nB=(18, 8);\nC=(24, 0... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "6303",
"split": "train"
} |
math | [
{
"content": "What is the value of $25_{10}+36_{10}$ in base 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2021_3",
"style": "rule"
} | {
"index": "6304",
"split": "train"
} |
math | [
{
"content": "In writing the integers from 20 through 99 inclusive, how many times is the digit 7 written? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "6305",
"split": "train"
} |
math | [
{
"content": "Among all the 5-digit numbers composed of the digits 1, 2, 3, 4, and 5 without repetition, how many are greater than 23145 and less than 43521? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "6306",
"split": "train"
} |
math | [
{
"content": "Given plane vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$ satisfy $\\langle \\overrightarrow{a}, \\overrightarrow{b} \\rangle = 60^\\circ$, and $\\{|\\overrightarrow{a}|, |\\overrightarrow{b}|, |\\overrightarrow{c}|\\} = \\{1, 2, 3\\}$, calculate the maximum value of ... | {
"ground_truth": "\\sqrt{7}+3",
"style": "rule"
} | {
"index": "6307",
"split": "train"
} |
math | [
{
"content": "Technology changes the world. Express sorting robots have become popular from Weibo to Moments. According to the introduction, these robots can not only automatically plan the optimal route, accurately place packages in the corresponding compartments, sense and avoid obstacles, automatically retur... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "6308",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$ , point $D$ lies on side $AC$ such that $\\angle ABD=\\angle C$ . Point $E$ lies on side $AB$ such that $BE=DE$ . $M$ is the midpoint of segment $CD$ . Point $H$ is the foot of the perpendicular from $A$ to $DE$ . Given $AH=2-\\sqrt{3}$ and $AB=1$ , find t... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "6309",
"split": "train"
} |
math | [
{
"content": "The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 5000? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8192",
"style": "rule"
} | {
"index": "6310",
"split": "train"
} |
math | [
{
"content": "A point $E$ on side $CD$ of a rectangle $ABCD$ is such that $\\triangle DBE$ is isosceles and $\\triangle ABE$ is right-angled. Find the ratio between the side lengths of the rectangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{\\sqrt{5} - 2}",
"style": "rule"
} | {
"index": "6311",
"split": "train"
} |
math | [
{
"content": "Isosceles right triangle $PQR$ (with $\\angle PQR = \\angle PRQ = 45^\\circ$ and hypotenuse $\\overline{PQ}$) encloses a right triangle $ABC$ (hypotenuse $\\overline{AB}$) as shown. Given $PC = 5$ and $BP = CQ = 4$, compute $AQ$. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "\\frac{5}{\\sqrt{2}}",
"style": "rule"
} | {
"index": "6312",
"split": "train"
} |
math | [
{
"content": "Using the numbers 0, 1, 2, 3, 4, 5 to form unique three-digit numbers, determine the total number of even numbers that can be formed. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "6313",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be five-digit palindromes (without leading zeroes) such that $a<b$ and there are no other five-digit palindromes strictly between $a$ and $b$. What are all possible values of $b-a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100, 110, 11",
"style": "rule"
} | {
"index": "6314",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\log _{3}(9x) \\cdot \\log _{3}(3x)$, where $\\frac {1}{9}\\leqslant x\\leqslant 9$.\n(I) If $m=\\log _{3}x$, find the range of $m$;\n(II) Find the maximum and minimum values of $f(x)$, and provide the corresponding $x$ values. Please output the final answer within \\boxe... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "6315",
"split": "train"
} |
math | [
{
"content": "How many 5-digit numbers $\\overline{a b c d e}$ exist such that digits $b$ and $d$ are each the sum of the digits to their immediate left and right? (That is, $b=a+c$ and $d=c+e$.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "330",
"style": "rule"
} | {
"index": "6316",
"split": "train"
} |
math | [
{
"content": "If\n\\[\\frac{\\sin x}{\\cos y} + \\frac{\\sin y}{\\cos x} = 1 \\quad \\text{and} \\quad \\frac{\\cos x}{\\sin y} + \\frac{\\cos y}{\\sin x} = 6,\\]then find $\\frac{\\tan x}{\\tan y} + \\frac{\\tan y}{\\tan x}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{124}{13}",
"style": "rule"
} | {
"index": "6317",
"split": "train"
} |
math | [
{
"content": "Phil rolls 5 fair 10-sided dice. What is the probability that at least two dice show the same number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1744}{2500}",
"style": "rule"
} | {
"index": "6318",
"split": "train"
} |
math | [
{
"content": "Let \\( f(x) \\) be a function defined on \\( \\mathbf{R} \\). If \\( f(x) + x^{2} \\) is an odd function, and \\( f(x) + 2^{x} \\) is an even function, then the value of \\( f(1) \\) is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{7}{4}",
"style": "rule"
} | {
"index": "6319",
"split": "train"
} |
math | [
{
"content": "Sita and Geeta are two sisters. If Sita's age is written after Geeta's age a four digit perfect square (number) is obtained. If the same exercise is repeated after 13 years another four digit perfect square (number) will be obtained. What is the sum of the present ages of Sita and Geeta? Please ou... | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "6320",
"split": "train"
} |
math | [
{
"content": "What is the ratio of the area of a regular hexagon inscribed in an equilateral triangle with side length $s$ to the area of a regular hexagon inscribed in a circle with radius $r$? Assume the height of the equilateral triangle equals the diameter of the circle, thus $s = r \\sqrt{3}$. Please outpu... | {
"ground_truth": "\\dfrac{9}{16}",
"style": "rule"
} | {
"index": "6321",
"split": "train"
} |
math | [
{
"content": "For each positive integer \\( n \\), let \\( \\varphi(n) \\) be the number of positive integers from 1 to \\( n \\) that are relatively prime to \\( n \\). Evaluate:\n$$\n\\sum_{n=1}^{\\infty} \\frac{\\varphi(n) 4^{n}}{7^{n}-4^{n}}\n$$ Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "28/9",
"style": "rule"
} | {
"index": "6322",
"split": "train"
} |
math | [
{
"content": "Point \\( D \\) lies on side \\( CB \\) of right triangle \\( ABC \\left(\\angle C = 90^{\\circ} \\right) \\), such that \\( AB = 5 \\), \\(\\angle ADC = \\arccos \\frac{1}{\\sqrt{10}}, DB = \\frac{4 \\sqrt{10}}{3} \\). Find the area of triangle \\( ABC \\). Please output the final answer within \... | {
"ground_truth": "15/4",
"style": "rule"
} | {
"index": "6323",
"split": "train"
} |
math | [
{
"content": "Given eight cubes with volumes \\(1, 8, 27, 64, 125, 216, 343,\\) and \\(512\\) cubic units, and each cube having its volume decreasing upwards, determine the overall external surface area of this arrangement. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1021",
"style": "rule"
} | {
"index": "6324",
"split": "train"
} |
math | [
{
"content": "For which values of \\(a\\) does the equation \\(|x-3| = a x - 1\\) have two solutions? Enter the midpoint of the interval of parameter \\(a\\) in the provided field. Round the answer to three significant digits according to rounding rules and enter it in the provided field. Please output the fina... | {
"ground_truth": "0.667",
"style": "rule"
} | {
"index": "6325",
"split": "train"
} |
math | [
{
"content": "Let $A B C D$ be a unit square. A circle with radius $\\frac{32}{49}$ passes through point $D$ and is tangent to side $A B$ at point $E$. Then $D E=\\frac{m}{n}$, where $m, n$ are positive integers and $\\operatorname{gcd}(m, n)=1$. Find $100 m+n$. Please output the final answer within \\boxed{}."... | {
"ground_truth": "807",
"style": "rule"
} | {
"index": "6326",
"split": "train"
} |
math | [
{
"content": "One hundred points labeled 1 to 100 are arranged in a $10 \\times 10$ grid such that adjacent points are one unit apart. The labels are increasing left to right, top to bottom (so the first row has labels 1 to 10 , the second row has labels 11 to 20, and so on). Convex polygon $\\mathcal{P}$ has t... | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "6327",
"split": "train"
} |
math | [
{
"content": "Given unit vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, and $\\overrightarrow{c}$ in the same plane, if the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is $60^{\\circ}$, find the maximum value of $(\\overrightarrow{a} - \\overrightarrow{b}) \\cdot (\\overrightarrow{a} - ... | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "6328",
"split": "train"
} |
math | [
{
"content": "Calculate $\\sqrt{75x} \\cdot \\sqrt{2x} \\cdot \\sqrt{14x}$ . Express your answer in simplest radical form in terms of $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10x \\sqrt{21x}",
"style": "rule"
} | {
"index": "6329",
"split": "train"
} |
math | [
{
"content": "Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "191",
"style": "rule"
} | {
"index": "6330",
"split": "train"
} |
math | [
{
"content": "The school plans to arrange 6 leaders to be on duty from May 1st to May 3rd. Each leader must be on duty for 1 day, with 2 leaders assigned each day. If leader A cannot be on duty on the 2nd, and leader B cannot be on duty on the 3rd, how many different methods are there to arrange the duty schedu... | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "6331",
"split": "train"
} |
math | [
{
"content": "Suppose \\(A, B\\) are the foci of a hyperbola and \\(C\\) is a point on the hyperbola. Given that the three sides of \\(\\triangle ABC\\) form an arithmetic sequence, and \\(\\angle ACB = 120^\\circ\\), determine the eccentricity of the hyperbola. Please output the final answer within \\boxed{}."... | {
"ground_truth": "7/2",
"style": "rule"
} | {
"index": "6332",
"split": "train"
} |
math | [
{
"content": "Given that $a$ is an odd multiple of $1183$, find the greatest common divisor of $2a^2+29a+65$ and $a+13$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "6333",
"split": "train"
} |
math | [
{
"content": "The fractional equation $\\dfrac{x-5}{x+2}=\\dfrac{m}{x+2}$ has a root, determine the value of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "6334",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ whose sum of the first $n$ terms is $S_n$, $a_1=15$, and it satisfies $\\frac{a_{n+1}}{2n-3} = \\frac{a_n}{2n-5}+1$, knowing $n$, $m\\in\\mathbb{N}$, and $n > m$, find the minimum value of $S_n - S_m$. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "-14",
"style": "rule"
} | {
"index": "6335",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=x^{3}+3ax^{2}+bx+a^{2}$ has an extreme value of $0$ at $x=-1$, find the value of $a-b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "6336",
"split": "train"
} |
math | [
{
"content": "Six small circles, each of radius $3$ units, are tangent to a large circle as shown. Each small circle also is tangent to its two neighboring small circles. What is the diameter of the large circle in units? [asy]\ndraw(Circle((-2,0),1));\ndraw(Circle((2,0),1));\ndraw(Circle((-1,1.73205081),1));\n... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "6337",
"split": "train"
} |
math | [
{
"content": "Write 22 as the sum of several distinct natural numbers so that the product of these numbers is maximized. The maximum product is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1008",
"style": "rule"
} | {
"index": "6338",
"split": "train"
} |
math | [
{
"content": "If a sequence's sum of the first $n$ terms, $S_n = 1 - 5 + 9 - \\ldots + (-1)^{n+1}(4n - 3)$ for $n \\in \\mathbb{N}^*$, find the value of $S_{15} - S_{22} + S_{31}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "134",
"style": "rule"
} | {
"index": "6339",
"split": "train"
} |
math | [
{
"content": "Given triangle \\( \\triangle ABC \\) with internal angles \\( A, B, C \\) opposite to sides \\( a, b, c \\) respectively, vectors \\( \\vec{m} = (1, 1 - \\sqrt{3} \\sin A) \\) and \\( \\vec{n} = (\\cos A, 1) \\), and \\( \\vec{m} \\perp \\vec{n} \\):\n(1) Find angle \\( A \\);\n(2) Given \\( b + ... | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "6340",
"split": "train"
} |
math | [
{
"content": "The altitude drawn to the base of an isosceles triangle is $8$, and the perimeter $32$. The area of the triangle is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "6341",
"split": "train"
} |
math | [
{
"content": "A triangle \\(ABC\\) is to be constructed so that \\(A\\) is at \\((3,2)\\), \\(B\\) is on the line \\(y=x\\), and \\(C\\) is on the \\(x\\)-axis. Find the minimum possible perimeter of \\(\\triangle ABC\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{26}",
"style": "rule"
} | {
"index": "6342",
"split": "train"
} |
math | [
{
"content": "Given the line $y=-x+1$ and the ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, which intersect at points $A$ and $B$.\n\n(1) If the eccentricity of the ellipse is $\\frac{\\sqrt{3}}{3}$ and the focal length is $2$, find the length of the line segment $AB$.\n\n(2) If vectors $\\... | {
"ground_truth": "\\sqrt{6}",
"style": "rule"
} | {
"index": "6343",
"split": "train"
} |
math | [
{
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $E: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$, and point $M$ is on $E$ with $MF_1$ perpendicular to the x-axis and $\\sin \\angle MF_2F_1 = \\frac{1}{3}$, find the eccentricity of $E$. Please output the final answer within \\bo... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "6344",
"split": "train"
} |
math | [
{
"content": "Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100 q+p$ is a perfect square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "179",
"style": "rule"
} | {
"index": "6345",
"split": "train"
} |
math | [
{
"content": "If $A$ and $B$ are nonzero digits, then the number of digits (not necessarily different) in the sum of the three whole numbers is\n$\\begin{array}{cccc} 9 & 8 & 7 & 6 \\\\ & A & 3 & 2 \\\\ & B & 1 \\\\ \\hline \\end{array}$ Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "6346",
"split": "train"
} |
math | [
{
"content": "Given that $\\alpha$ and $\\beta$ are acute angles, and $\\cos(\\alpha + \\beta) = \\frac{3}{5}$, $\\sin\\alpha = \\frac{5}{13}$, find the value of $\\cos\\beta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{56}{65}",
"style": "rule"
} | {
"index": "6347",
"split": "train"
} |
math | [
{
"content": "Arrange 7 staff members to be on duty from May 1st to May 7th. Each person works for one day, among them, person A and person B are not scheduled on May 1st and 2nd. The total number of different arrangements is $\\boxed{\\text{\\_\\_\\_\\_\\_\\_\\_\\_}}$. Please output the final answer within \\b... | {
"ground_truth": "2400",
"style": "rule"
} | {
"index": "6348",
"split": "train"
} |
math | [
{
"content": "The triangle $ABC$ is isosceles with $AB=AC$, and $\\angle{BAC}<60^{\\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\\angle{ABD}\\equiv\\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\\angle{BDC}$ and $\\angle{... | {
"ground_truth": "120^\\circ",
"style": "rule"
} | {
"index": "6349",
"split": "train"
} |
math | [
{
"content": "A path of length $n$ is a sequence of points $\\left(x_{1}, y_{1}\\right),\\left(x_{2}, y_{2}\\right), \\ldots,\\left(x_{n}, y_{n}\\right)$ with integer coordinates such that for all $i$ between 1 and $n-1$ inclusive, either (1) $x_{i+1}=x_{i}+1$ and $y_{i+1}=y_{i}$ (in which case we say the $i$th... | {
"ground_truth": "1024",
"style": "rule"
} | {
"index": "6350",
"split": "train"
} |
math | [
{
"content": "Canadian currency has coins with values $2.00, 1.00, 0.25, 0.10,$ and $0.05. Barry has 12 coins including at least one of each of these coins. Find the smallest total amount of money that Barry could have. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3.75",
"style": "rule"
} | {
"index": "6351",
"split": "train"
} |
math | [
{
"content": "The game of Penta is played with teams of five players each, and there are five roles the players can play. Each of the five players chooses two of five roles they wish to play. If each player chooses their roles randomly, what is the probability that each role will have exactly two players? Pleas... | {
"ground_truth": "\\frac{51}{2500}",
"style": "rule"
} | {
"index": "6352",
"split": "train"
} |
math | [
{
"content": "Given two functions $f(x) = e^{2x-3}$ and $g(x) = \\frac{1}{4} + \\ln \\frac{x}{2}$, if $f(m) = g(n)$ holds, calculate the minimum value of $n-m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2} + \\ln 2",
"style": "rule"
} | {
"index": "6353",
"split": "train"
} |
math | [
{
"content": "Convert the binary number $1101100_{(2)}$ to a decimal number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "6354",
"split": "train"
} |
math | [
{
"content": "The Crystal Products Factory in the East Sea had an annual output of 100,000 pieces last year, with each crystal product selling for 100 yuan and a fixed cost of 80 yuan. Starting this year, the factory invested 1 million yuan in technology costs and plans to invest an additional 1 million yuan in... | {
"ground_truth": "520",
"style": "rule"
} | {
"index": "6355",
"split": "train"
} |
math | [
{
"content": "Apartment rentals in Fairview run $\\$$0.90 per square foot. Jillian has determined that she can afford $\\$$630 per month for rent. What is the largest apartment, in square feet, she should consider at the given rate? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "700",
"style": "rule"
} | {
"index": "6356",
"split": "train"
} |
math | [
{
"content": "The equations $x^3 + Ax + 10 = 0$ and $x^3 + Bx^2 + 50 = 0$ have two roots in common. Then the product of these common roots can be expressed in the form $a \\sqrt[b]{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified. Find $a + b + c.$ Please output the final answer within \\bo... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "6357",
"split": "train"
} |
math | [
{
"content": "Find the length of \\(PQ\\) in the triangle below, where \\(PQR\\) is a right triangle with \\( \\angle RPQ = 45^\\circ \\) and the length \\(PR\\) is \\(10\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10\\sqrt{2}",
"style": "rule"
} | {
"index": "6358",
"split": "train"
} |
math | [
{
"content": "The distance from Stockholm to Malmö on a map is 120 cm. The scale on the map is 1 cm: 20 km. If there is a stop in between at Lund, which is 30 cm away from Malmö on the same map, how far is it from Stockholm to Malmö passing through Lund, in kilometers? Please output the final answer within \\bo... | {
"ground_truth": "2400",
"style": "rule"
} | {
"index": "6359",
"split": "train"
} |
math | [
{
"content": "A metro network has at least 4 stations on each line, with no more than three transfer stations per line. No transfer station has more than two lines crossing. What is the maximum number of lines such a network can have if it is possible to travel from any station to any other station with no more... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "6360",
"split": "train"
} |
math | [
{
"content": "A three-dimensional rectangular box with dimensions $X$, $Y$, and $Z$ has faces whose surface areas are $24$, $24$, $48$, $48$, $72$, and $72$ square units. What is $X$ + $Y$ + $Z$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "6361",
"split": "train"
} |
math | [
{
"content": "Find the number of real solutions to the equation:\n\\[\n\\frac{1^2}{x - 1} + \\frac{2^2}{x - 2} + \\frac{3^2}{x - 3} + \\dots + \\frac{120^2}{x - 120} = x.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "121",
"style": "rule"
} | {
"index": "6362",
"split": "train"
} |
math | [
{
"content": "In a regular hexagon \\( ABCDEF \\), the diagonals \\( AC \\) and \\( CE \\) are divided by interior points \\( M \\) and \\( N \\) in the following ratio: \\( AM : AC = CN : CE = r \\). If the points \\( B, M, N \\) are collinear, find the ratio \\( r \\). Please output the final answer within \\... | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "6363",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ $c,$ $x,$ $y,$ $z$ be nonzero complex numbers such that\n\\[a = \\frac{b + c}{x - 2}, \\quad b = \\frac{a + c}{y - 2}, \\quad c = \\frac{a + b}{z - 2},\\]and $xy + xz + yz = 5$ and $x + y + z = 3,$ find $xyz.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "6364",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin(x+\\pi) + \\cos(x-\\pi) = \\frac{1}{2}, x \\in (0, \\pi)$.\n(1) Find the value of $\\sin x \\cos x$;\n(2) Find the value of $\\sin x - \\cos x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{7}}{2}",
"style": "rule"
} | {
"index": "6365",
"split": "train"
} |
math | [
{
"content": "If $x - y = 12$ and $x + y = 6$, what is the value of $y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "6366",
"split": "train"
} |
math | [
{
"content": "If $h(x)$ is a function whose domain is $[-8,8]$, and $g(x)=h\\left(\\frac x2\\right)$, then the domain of $g(x)$ is an interval of what width? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "6367",
"split": "train"
} |
math | [
{
"content": "Assume integers \\( u \\) and \\( v \\) satisfy \\( 0 < v < u \\), and let \\( A \\) be \\((u, v)\\). Points are defined as follows: \\( B \\) is the reflection of \\( A \\) over the line \\( y = x \\), \\( C \\) is the reflection of \\( B \\) over the \\( y \\)-axis, \\( D \\) is the reflection o... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "6368",
"split": "train"
} |
math | [
{
"content": "Define two binary operations on real numbers where $a \\otimes b = \\frac{a+b}{a-b}$ and $b \\oplus a = \\frac{b-a}{b+a}$. Compute the value of $(8\\otimes 6) \\oplus 2$.\nA) $\\frac{5}{9}$ \nB) $\\frac{7}{9}$ \nC) $\\frac{12}{9}$ \nD) $\\frac{1}{9}$ \nE) $\\frac{14}{9}$ Please output the fin... | {
"ground_truth": "\\frac{5}{9}",
"style": "rule"
} | {
"index": "6369",
"split": "train"
} |
math | [
{
"content": "Let us consider a set $S = \\{ a_1 < a_2 < \\ldots < a_{2004}\\}$ , satisfying the following properties: $f(a_i) < 2003$ and $f(a_i) = f(a_j) \\quad \\forall i, j$ from $\\{1, 2,\\ldots , 2004\\}$ , where $f(a_i)$ denotes number of elements which are relatively prime with $a_i$ . Find the... | {
"ground_truth": "1003",
"style": "rule"
} | {
"index": "6370",
"split": "train"
} |
math | [
{
"content": "Two equilateral triangles are contained in a square whose side length is $2\\sqrt{3}$. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "8\\sqrt{3} - 12",
"style": "rule"
} | {
"index": "6371",
"split": "train"
} |
math | [
{
"content": "Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest number possible of black edges is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "6372",
"split": "train"
} |
math | [
{
"content": "Let M be a subst of {1,2,...,2006} with the following property: For any three elements x,y and z (x<y<z) of M, x+y does not divide z. Determine the largest possible size of M. Justify your claim. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1004",
"style": "rule"
} | {
"index": "6373",
"split": "train"
} |
math | [
{
"content": "Each of the $25$ balls is tossed independently and at random into one of $5$ bins. Let $r$ be the probability that one bin ends up with $6$ balls, another with $7$ balls, and the other three with $4$ balls each. Let $s$ be the probability that one bin ends up with $5$ balls and the other four with... | {
"ground_truth": "\\frac{10 \\cdot \\binom{25}{6}\\binom{19}{7}\\binom{12}{4}\\binom{8}{4}\\binom{4}{4}}{\\binom{25}{5}\\binom{20}{5}\\binom{15}{5}\\binom{10}{5}\\binom{5}{5}}",
"style": "rule"
} | {
"index": "6374",
"split": "train"
} |
math | [
{
"content": "It is easy to place the complete set of ships for the game \"Battleship\" on a $10 \\times 10$ board (see illustration). What is the smallest square board on which this set can be placed? (Recall that according to the rules, ships must not touch each other, even at the corners.) Please output the ... | {
"ground_truth": "7 \\times 7",
"style": "rule"
} | {
"index": "6375",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ whose sum of the first $n$ terms is $S_n$, and the point $(n, \\frac{S_n}{n})$ lies on the line $y= \\frac{1}{2}x+ \\frac{11}{2}$. The sequence $\\{b_n\\}$ satisfies $b_{n+2}-2b_{n+1}+b_n=0$ $(n\\in{N}^*)$, and $b_3=11$, with the sum of the first $9$ terms being $153$.... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "6376",
"split": "train"
} |
math | [
{
"content": "What is the equation of the oblique asymptote of the graph of $\\frac{2x^2+7x+10}{2x+3}$?\n\nEnter your answer in the form $y = mx + b.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "y = x+2",
"style": "rule"
} | {
"index": "6377",
"split": "train"
} |
math | [
{
"content": "Point $O$ is the center of the regular octagon $ABCDEFGH$, and $Y$ is the midpoint of side $\\overline{CD}$. Calculate the fraction of the area of the octagon that is shaded, where the shaded region includes triangles $\\triangle DEO$, $\\triangle EFO$, $\\triangle FGO$, and half of $\\triangle DC... | {
"ground_truth": "\\frac{7}{16}",
"style": "rule"
} | {
"index": "6378",
"split": "train"
} |
math | [
{
"content": "Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\\ell$ passing through $B$ , we color red the foot of the perpendicular from $C$ to $\\ell$ . The set of red points enclose a bounded region of area $\\mathcal{A}$ . Find $\\lfloor \\mathcal{A} \\rfloor$ ... | {
"ground_truth": "157",
"style": "rule"
} | {
"index": "6379",
"split": "train"
} |
math | [
{
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-12.5, 12.5]\\). 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": "650",
"style": "rule"
} | {
"index": "6380",
"split": "train"
} |
math | [
{
"content": "A shepherd uses 15 segments of fencing, each 2 meters long, to form a square or rectangular sheep pen with one side against a wall. What is the maximum area of the sheep pen in square meters? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "6381",
"split": "train"
} |
math | [
{
"content": "The positive difference between the two roots of the quadratic equation $3x^2 - 7x - 8 = 0$ can be written as $\\frac{\\sqrt{m}}{n}$, where $n$ is an integer and $m$ is an integer not divisible by the square of any prime number. Find $m + n$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "148",
"style": "rule"
} | {
"index": "6382",
"split": "train"
} |
math | [
{
"content": "Given $\\sin\\left(\\frac{\\pi}{3} - \\alpha\\right) = \\frac{1}{3}$, calculate $\\sin\\left(\\frac{\\pi}{6} - 2\\alpha\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "- \\frac{7}{9}",
"style": "rule"
} | {
"index": "6383",
"split": "train"
} |
math | [
{
"content": "A courier set out on a moped from city $A$ to city $B$, which are 120 km apart. One hour later, a second courier on a motorcycle left from $A$. The second courier caught up to the first, delivered a message, and immediately returned to $A$ at the same speed, arriving back in $A$ at the same time t... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "6384",
"split": "train"
} |
math | [
{
"content": "Triangle $XYZ$ is a right, isosceles triangle. Angle $X$ measures 45 degrees. What is the number of degrees in the measure of angle $Y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "6385",
"split": "train"
} |
math | [
{
"content": "Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\\triangle ABC \\sim \\triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\\triangle ABC \\sim \\triangle CEB$ and the area of $\\triangle AED$ is $17$ times the area of $\\triangle CEB$. What is $\\tfrac{A... | {
"ground_truth": "2+\\sqrt{5}",
"style": "rule"
} | {
"index": "6386",
"split": "train"
} |
math | [
{
"content": "Given a triangular pyramid \\( P-ABC \\) with a base that is an equilateral triangle of side length \\( 4 \\sqrt{3} \\), and with \\( PA=3 \\), \\( PB=4 \\), and \\( PC=5 \\). If \\( O \\) is the center of the triangle \\( ABC \\), find the length of \\( PO \\). Please output the final answer with... | {
"ground_truth": "\\frac{\\sqrt{6}}{3}",
"style": "rule"
} | {
"index": "6387",
"split": "train"
} |
math | [
{
"content": "(Geometry Proof Exercise) From a point A outside a circle ⊙O with radius 2, draw a line intersecting ⊙O at points C and D. A line segment AB is tangent to ⊙O at B. Given that AC=4 and AB=$4 \\sqrt {2}$, find $tan∠DAB$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac { \\sqrt {2}}{4}",
"style": "rule"
} | {
"index": "6388",
"split": "train"
} |
math | [
{
"content": "Given that \\(\\theta\\) is an angle in the third quadrant, and \\(\\sin^{4}\\theta + \\cos^{4}\\theta = \\frac{5}{9}\\), determine the value of \\(\\sin 2\\theta\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "6389",
"split": "train"
} |
math | [
{
"content": "Given a circle $C$ with the equation $(x-a)^2 + (y-2)^2 = 4$ ($a > 0$) and a line $l$ with the equation $x - y + 3 = 0$, which intersects the circle at points $A$ and $B$:\n1. When the chord AB cut by line $l$ on circle $C$ has a length of $2\\sqrt{2}$, find the value of $a$.\n2. If there exists a... | {
"ground_truth": "2\\sqrt{2} - 1",
"style": "rule"
} | {
"index": "6390",
"split": "train"
} |
math | [
{
"content": "For some complex number $z$ with $|z| = 3,$ there is some real $\\lambda > 1$ such that $z,$ $z^2,$ and $\\lambda z$ form an equilateral triangle in the complex plane. Find $\\lambda.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1 + \\sqrt{33}}{2}",
"style": "rule"
} | {
"index": "6391",
"split": "train"
} |
math | [
{
"content": "Given: $A=2a^{2}-5ab+3b$, $B=4a^{2}+6ab+8a$. \n$(1)$ Simplify: $2A-B$; \n$(2)$ If $a=-2$, $b=1$, find the value of $2A-B$; \n$(3)$ If the value of the algebraic expression $2A-B$ is independent of $a$, find the value of $b$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "6392",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $C$: $\\dfrac{x^{2}}{a^{2}} + \\dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with its right focus at $F(\\sqrt{3}, 0)$, and point $M(-\\sqrt{3}, \\dfrac{1}{2})$ on ellipse $C$.\n\n(Ⅰ) Find the standard equation of ellipse $C$;\n\n(Ⅱ) Line $l$ passes through point $F$ and intersects ellip... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "6393",
"split": "train"
} |
math | [
{
"content": "Find the largest integer less than 74 that leaves a remainder of 3 when divided by 7. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "73",
"style": "rule"
} | {
"index": "6394",
"split": "train"
} |
math | [
{
"content": "Alex is thinking of a number that is divisible by all of the positive integers 1 through 200 inclusive except for two consecutive numbers. What is the smaller of these numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "6395",
"split": "train"
} |
math | [
{
"content": "Given $995 + 997 + 999 + 1001 + 1003 = 5100 - N$, determine $N$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "6396",
"split": "train"
} |
math | [
{
"content": "The product of three different positive integers is equal to $7^3$. What is the sum of the three integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "6397",
"split": "train"
} |
math | [
{
"content": "We build a $4 \\times 4 \\times 4$ cube out of sugar cubes. How many different rectangular parallelepipeds can the sugar cubes determine, if the rectangular parallelepipeds differ in at least one sugar cube? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": "6398",
"split": "train"
} |
math | [
{
"content": "Vasya wrote consecutive natural numbers \\( N \\), \\( N+1 \\), \\( N+2 \\), and \\( N+3 \\) in rectangles. Under each rectangle, he wrote the sum of the digits of the corresponding number in a circle.\n\nThe sum of the numbers in the first two circles turned out to be 200, and the sum of the numb... | {
"ground_truth": "103",
"style": "rule"
} | {
"index": "6399",
"split": "train"
} |
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