data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "If $y=a+\\frac{b}{x}$, where $a$ and $b$ are constants, and if $y=1$ when $x=-1$, and $y=5$ when $x=-5$, then $a+b$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "37200",
"split": "train"
} |
math | [
{
"content": "Car A and Car B are traveling in opposite directions on a road parallel to a railway. A 180-meter-long train is moving in the same direction as Car A at a speed of 60 km/h. The time from when the train catches up with Car A until it meets Car B is 5 minutes. If it takes the train 30 seconds to com... | {
"ground_truth": "1.25",
"style": "rule"
} | {
"index": "37201",
"split": "train"
} |
math | [
{
"content": "A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, t... | {
"ground_truth": "448",
"style": "rule"
} | {
"index": "37202",
"split": "train"
} |
math | [
{
"content": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a function satisfying $f(x) f(y)=f(x-y)$. Find all possible values of $f(2017)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0, 1",
"style": "rule"
} | {
"index": "37203",
"split": "train"
} |
math | [
{
"content": "Given that $f(x)$ is an odd function on $\\mathbb{R}$, when $x\\geqslant 0$, $f(x)= \\begin{cases} \\log _{\\frac {1}{2}}(x+1),0\\leqslant x < 1 \\\\ 1-|x-3|,x\\geqslant 1\\end{cases}$. Find the sum of all the zeros of the function $y=f(x)+\\frac {1}{2}$. Please output the final answer within \\bo... | {
"ground_truth": "\\sqrt {2}-1",
"style": "rule"
} | {
"index": "37204",
"split": "train"
} |
math | [
{
"content": "Given $m\\in R$, determine the condition for the lines $l_{1}$: $mx+2y-1=0$ and $l_{2}$: $3x+\\left(m+1\\right)y+1=0$ to be perpendicular. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2}{5}",
"style": "rule"
} | {
"index": "37205",
"split": "train"
} |
math | [
{
"content": "What is the value of $12345 + 23451 + 34512 + 45123 + 51234$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "166665",
"style": "rule"
} | {
"index": "37206",
"split": "train"
} |
math | [
{
"content": "The edge of cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is equal to 1. Construct a cross-section of the cube by a plane that has the maximum perimeter. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{2}",
"style": "rule"
} | {
"index": "37207",
"split": "train"
} |
math | [
{
"content": "Three cars start simultaneously from City A, heading towards City B along the same highway. The second car travels 4 kilometers per hour less than the first car and 6 kilometers per hour more than the third car. The second car arrives at City B 3 minutes later than the first car and 5 minutes earl... | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "37208",
"split": "train"
} |
math | [
{
"content": "Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "950",
"style": "rule"
} | {
"index": "37209",
"split": "train"
} |
math | [
{
"content": "If $a, b, c>0$, what is the smallest possible value of $\\left\\lfloor\\frac{a+b}{c}\\right\\rfloor+\\left\\lfloor\\frac{b+c}{a}\\right\\rfloor+\\left\\lfloor\\frac{c+a}{b}\\right\\rfloor$? (Note that $\\lfloor x\\rfloor$ denotes the greatest integer less than or equal to $x$.) Please output the f... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "37210",
"split": "train"
} |
math | [
{
"content": "Find an ordered pair $(x,y)$ that satisfies both of the equations below: \\begin{align*} 2x - 3y &= -5,\\\\ 5x - 2y &= 4. \\end{align*} Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(2,3)",
"style": "rule"
} | {
"index": "37211",
"split": "train"
} |
math | [
{
"content": "How many multiples of 5 are there between 105 and 500? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "79",
"style": "rule"
} | {
"index": "37212",
"split": "train"
} |
math | [
{
"content": "Find the matrix $\\mathbf{M}$ such that\n\\[\\mathbf{M} \\begin{pmatrix} -3 & 4 & 0 \\\\ 5 & -7 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} = \\mathbf{I}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} -7 & -4 & 0 \\\\ -5 & -3 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix}",
"style": "rule"
} | {
"index": "37213",
"split": "train"
} |
math | [
{
"content": "Let the sequence of non-negative integers $\\left\\{a_{n}\\right\\}$ satisfy:\n$$\na_{n} \\leqslant n \\quad (n \\geqslant 1), \\quad \\text{and} \\quad \\sum_{k=1}^{n-1} \\cos \\frac{\\pi a_{k}}{n} = 0 \\quad (n \\geqslant 2).\n$$\n\nFind all possible values of $a_{2021}$. Please output the final... | {
"ground_truth": "2021",
"style": "rule"
} | {
"index": "37214",
"split": "train"
} |
math | [
{
"content": "Given the function \\( y = \\sqrt{2x^2 + 2} \\) with its graph represented as curve \\( G \\), and the focus of curve \\( G \\) denoted as \\( F \\), two lines \\( l_1 \\) and \\( l_2 \\) pass through \\( F \\) and intersect curve \\( G \\) at points \\( A, C \\) and \\( B, D \\) respectively, suc... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "37215",
"split": "train"
} |
math | [
{
"content": "Given a tetrahedron \\( S-ABC \\), point \\( A_1 \\) is the centroid of \\( \\triangle SBC \\). Point \\( G \\) is on the segment \\( AA_1 \\) such that \\(\\frac{|AG|}{|GA_1|}=3\\). Line \\( SG \\) intersects the plane of \\( \\triangle ABC \\) at point \\( M \\). Determine \\(\\frac{|A_1 M|}{|AS... | {
"ground_truth": "1/3",
"style": "rule"
} | {
"index": "37216",
"split": "train"
} |
math | [
{
"content": "An even perfect square in the decimal system is of the form: $\\overline{a b 1 a b}$. What is this perfect square? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "76176",
"style": "rule"
} | {
"index": "37217",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=x+2$ and $g(x)=x/3.$ Also denote the inverses to these functions as $f^{-1}$ and $g^{-1}.$ Compute \\[f(g^{-1}(f^{-1}(f^{-1}(g(f(19)))))).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "37218",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression $\\dfrac{\\sqrt[6]{5}}{\\sqrt[4]{5}}$. What power of 5 does this expression represent? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{12}",
"style": "rule"
} | {
"index": "37219",
"split": "train"
} |
math | [
{
"content": "( Dick Gibbs ) For a given positive integer $k$ find, in terms of $k$ , the minimum value of $N$ for which there is a set of $2k+1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $N/2$ . Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "\\[ N = 2k^3 + 3k^2 + 3k \\]",
"style": "rule"
} | {
"index": "37220",
"split": "train"
} |
math | [
{
"content": "The integer 2014 is between which powers of 10? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10^{3} \\text{ and } 10^{4}",
"style": "rule"
} | {
"index": "37221",
"split": "train"
} |
math | [
{
"content": "Determine the area enclosed by the parabola $y = x^{2} - 5x + 6$ and the coordinate axes (and adjacent to both axes). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.666666666666667",
"style": "rule"
} | {
"index": "37222",
"split": "train"
} |
math | [
{
"content": "What is the sum of all integer solutions to $1<(x-2)^2<25$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "37223",
"split": "train"
} |
math | [
{
"content": "Two friends agree to meet at a specific place between 12:00 PM and 12:30 PM. The first one to arrive waits for the other for 20 minutes before leaving. Find the probability that the friends will meet, assuming each chooses their arrival time randomly (between 12:00 PM and 12:30 PM) and independent... | {
"ground_truth": "8/9",
"style": "rule"
} | {
"index": "37224",
"split": "train"
} |
math | [
{
"content": "A triangular lattice is formed with seven points arranged as follows: six points form a regular hexagon and one point is at the center. Each point is one unit away from its nearest neighbor. Determine how many equilateral triangles can be formed where all vertices are on this lattice. Assume the p... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "37225",
"split": "train"
} |
math | [
{
"content": "How many ways can a schedule of 4 mathematics courses - algebra, geometry, number theory, and calculus - be created in an 8-period day if exactly one pair of these courses can be taken in consecutive periods, and the other courses must not be consecutive? Please output the final answer within \\bo... | {
"ground_truth": "1680",
"style": "rule"
} | {
"index": "37226",
"split": "train"
} |
math | [
{
"content": "Given a random variable $ξ$ follows a normal distribution $N(2,σ^{2})$, and $P(ξ \\leqslant 4-a) = P(ξ \\geqslant 2+3a)$, solve for the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "37227",
"split": "train"
} |
math | [
{
"content": "For a constant $c,$ in spherical coordinates $(\\rho,\\theta,\\phi),$ find the shape described by the equation\n\\[\\phi = c.\\](A) Line\n(B) Circle\n(C) Plane\n(D) Sphere\n(E) Cylinder\n(F) Cone\n\nEnter the letter of the correct option. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "\\text{(F)}",
"style": "rule"
} | {
"index": "37228",
"split": "train"
} |
math | [
{
"content": "Given the symbol $R_k$ represents an integer whose base-ten representation is a sequence of $k$ ones, find the number of zeros in the quotient $Q=R_{28}/R_8$ when $R_{28}$ is divided by $R_8$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "37229",
"split": "train"
} |
math | [
{
"content": "Assume every 7-digit whole number is a possible telephone number except those that begin with a digit less than 3. What fraction of telephone numbers begin with $9$ and have $3$ as their middle digit (i.e., fourth digit)?\nA) $\\frac{1}{60}$\nB) $\\frac{1}{70}$\nC) $\\frac{1}{80}$\nD) $\\frac{1}{9... | {
"ground_truth": "\\frac{1}{70}",
"style": "rule"
} | {
"index": "37230",
"split": "train"
} |
math | [
{
"content": "Compute $\\oplus(\\oplus(2,4,5), \\oplus(3,5,4), \\oplus(4,2,5))$, where $\\oplus(x,y,z) = \\frac{x+y-z}{y-z}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10}{13}",
"style": "rule"
} | {
"index": "37231",
"split": "train"
} |
math | [
{
"content": "The equations of the asymptotes of the hyperbola \\\\( \\frac {x^{2}}{3}- \\frac {y^{2}}{6}=1 \\\\) are \\_\\_\\_\\_\\_\\_, and its eccentricity is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt {3}",
"style": "rule"
} | {
"index": "37232",
"split": "train"
} |
math | [
{
"content": "In how many ways can 7 people be seated in a row of chairs if two of the people, Wilma and Paul, refuse to sit next to each other? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3600",
"style": "rule"
} | {
"index": "37233",
"split": "train"
} |
math | [
{
"content": "For any two positive integers m and n, define an operation \"※\" as follows: when m and n are both positive even numbers or positive odd numbers, m※n=m+n; when one of m and n is a positive even number and the other is a positive odd number, m※n=mn. Determine the number of elements in the set M={(a... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "37234",
"split": "train"
} |
math | [
{
"content": "Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaini... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "37235",
"split": "train"
} |
math | [
{
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a\\_n\\}$ is $S\\_n$, where the first term $a\\_1 > 0$ and the common difference $d < 0$. For any $n \\in \\mathbb{N}^*$, there exists $k \\in \\mathbb{N}^*$ such that $a\\_k = S\\_n$. Find the minimum value of $k - 2n$. Please output th... | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "37236",
"split": "train"
} |
math | [
{
"content": "Find all triples $(x, y, z)$ of nonnegative integers such that\n$$ x^5+x^4+1=3^y7^z $$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(0, 0, 0), (1, 1, 0), (2, 0, 2)",
"style": "rule"
} | {
"index": "37237",
"split": "train"
} |
math | [
{
"content": "Consider all angles measured in degrees. Calculate the product $\\prod_{k=1}^{22} \\csc^2(4k-3)^\\circ = m^n$, where $m$ and $n$ are integers greater than 1. Find the value of $m+n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "37238",
"split": "train"
} |
math | [
{
"content": "Five integers are written on a board. By summing them in pairs, the following set of 10 numbers was obtained: $-1, 5, 8, 9, 11, 12, 14, 18, 20, 24$. Determine which numbers are written on the board. Provide their product as the answer. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "-2002",
"style": "rule"
} | {
"index": "37239",
"split": "train"
} |
math | [
{
"content": "Given that a five-digit palindromic number is equal to the product of 45 and a four-digit palindromic number (i.e., $\\overline{\\mathrm{abcba}} = 45 \\times \\overline{\\text{deed}}$), find the largest possible value of the five-digit palindromic number. Please output the final answer within \\bo... | {
"ground_truth": "59895",
"style": "rule"
} | {
"index": "37240",
"split": "train"
} |
math | [
{
"content": "Maryam has a fair tetrahedral die, with the four faces of the die labeled 1 through 4. At each step, she rolls the die and records which number is on the bottom face. She stops when the current number is greater than or equal to the previous number. (In particular, she takes at least two steps.)... | {
"ground_truth": "625/256",
"style": "rule"
} | {
"index": "37241",
"split": "train"
} |
math | [
{
"content": "For what value of $n$ does $|6 + ni| = 6\\sqrt{5}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "37242",
"split": "train"
} |
math | [
{
"content": "Point $P$ is a moving point on the curve $C_{1}$: $(x-2)^{2}+y^{2}=4$. Taking the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. Point $P$ is rotated counterclockwise by $90^{\\circ}$ around the pole $O$ to obtain point $Q$. Su... | {
"ground_truth": "3- \\sqrt {3}",
"style": "rule"
} | {
"index": "37243",
"split": "train"
} |
math | [
{
"content": "A certain unit decides to invest $3200$ yuan to build a warehouse (in the shape of a rectangular prism) with a constant height. The back wall will be built reusing the old wall at no cost, the front will be made of iron grilles at a cost of $40$ yuan per meter in length, and the two side walls wil... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "37244",
"split": "train"
} |
math | [
{
"content": "For what values of the constant $c$ does the graph of $f(x) = \\frac{x^2-x+c}{x^2+x-6}$ have exactly one vertical asymptote?\n\nEnter the possible values of $c,$ separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2 \\text{ or } -12",
"style": "rule"
} | {
"index": "37245",
"split": "train"
} |
math | [
{
"content": "Simplify and then evaluate the expression: $$(1- \\frac {1}{a-2})÷ \\frac {a^{2}-6a+9}{2a-4}$$, where $$a=2 \\sqrt {3}+3$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "37246",
"split": "train"
} |
math | [
{
"content": "【Information Extraction】 \nIn some cases, it is not necessary to calculate the result to remove the absolute value symbol. For example: $|6+7|=6+7$, $|6-7|=7-6$, $|7-6|=7-6$, $|-6-7|=6+7$. \n【Initial Experience】 \n$(1)$ According to the above rules, write the following expressions in the form w... | {
"ground_truth": "\\frac{505}{1011}",
"style": "rule"
} | {
"index": "37247",
"split": "train"
} |
math | [
{
"content": "Simplify $(3p^3 - 5p + 6) + (4 - 6p^2 + 2p)$. Express your answer in the form $Ap^3 + Bp^2 + Cp +D$, where $A$, $B$, $C$, and $D$ are numbers (possibly negative). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3p^3 - 6p^2 - 3p + 10",
"style": "rule"
} | {
"index": "37248",
"split": "train"
} |
math | [
{
"content": "Fill in the blanks:\n\n(1) In $\\triangle ABC$, $A=60^{\\circ}$, $b = 1$, and the area of $\\triangle ABC$ is $\\sqrt{3}$. Find the value of $\\dfrac{a+b+c}{\\sin A+\\sin B+\\sin C}$.\n\n(2) In an arithmetic sequence with a common difference not equal to $0$, ${a_1}+{a_3}=8$, and $a_4$ is the geom... | {
"ground_truth": "765",
"style": "rule"
} | {
"index": "37249",
"split": "train"
} |
math | [
{
"content": "If \\(\\cos ^{4} \\theta + \\sin ^{4} \\theta + (\\cos \\theta \\cdot \\sin \\theta)^{4} + \\frac{1}{\\cos ^{4} \\theta + \\sin ^{4} \\theta} = \\frac{41}{16}\\), find the value of \\(\\sin ^{2} \\theta\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "37250",
"split": "train"
} |
math | [
{
"content": "A portion of the graph of $y = f(x)$ is shown in red below, where $f(x)$ is a quadratic function. The distance between grid lines is $1$ unit.\n\nWhat is the sum of all distinct numbers $x$ such that $f(f(f(x)))=-3$ ?\n\n[asy]\n\nsize(150);\n\nreal ticklen=3;\n\nreal tickspace=2;\n\nreal tickleng... | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "37251",
"split": "train"
} |
math | [
{
"content": "The squares of a $3 \\times 3$ grid are filled with positive integers such that 1 is the label of the upperleftmost square, 2009 is the label of the lower-rightmost square, and the label of each square divides the one directly to the right of it and the one directly below it. How many such labelin... | {
"ground_truth": "2448",
"style": "rule"
} | {
"index": "37252",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural number that can be represented in exactly two ways as \\(3x + 4y\\), where \\(x\\) and \\(y\\) are natural numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "37253",
"split": "train"
} |
math | [
{
"content": "Write $x^{10} + x^5 + 1$ as the product of two polynomials with integer coefficients. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(x^2 + x + 1)(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)",
"style": "rule"
} | {
"index": "37254",
"split": "train"
} |
math | [
{
"content": "Distribute 61 books to a class of students. If at least one person must receive at least 3 books, what is the maximum number of students in the class? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "37255",
"split": "train"
} |
math | [
{
"content": "A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip? Please output the f... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "37256",
"split": "train"
} |
math | [
{
"content": "The base of the pyramid \\( P A B C D \\) is a trapezoid \\( A B C D \\), with the base \\( A D \\) being twice as large as the base \\( B C \\). The segment \\( M N \\) is the midline of the triangle \\( A B P \\), parallel to the side \\( A B \\). Find the ratio of the volumes of the two solids ... | {
"ground_truth": "\\frac{5}{13}",
"style": "rule"
} | {
"index": "37257",
"split": "train"
} |
math | [
{
"content": "Set $A$ has $20$ elements, and set $B$ has $15$ elements. What is the smallest possible number of elements in $A \\cup B$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "37258",
"split": "train"
} |
math | [
{
"content": "The greatest common divisor of 30 and some number between 70 and 90 is 6. What is the number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "37259",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), let F be the right focus of the hyperbola. A perpendicular line from point F to the x-axis intersects the two asymptotes at points A and B, and intersects the hyperbola in the first quadrant at point P. Let O be the or... | {
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "37260",
"split": "train"
} |
math | [
{
"content": "A hotel has three types of rooms available: a triple room, a double room, and a single room (each type of room can only accommodate the corresponding number of people). There are 4 adult men and 2 little boys looking for accommodation. The little boys should not stay in a room by themselves (they ... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "37261",
"split": "train"
} |
math | [
{
"content": "Find the sum: \\( S = 19 \\cdot 20 \\cdot 21 + 20 \\cdot 21 \\cdot 22 + \\cdots + 1999 \\cdot 2000 \\cdot 2001 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6 \\left( \\binom{2002}{4} - \\binom{21}{4} \\right)",
"style": "rule"
} | {
"index": "37262",
"split": "train"
} |
math | [
{
"content": "How many natural numbers between 200 and 400 are divisible by 8? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "37263",
"split": "train"
} |
math | [
{
"content": "What is the total number of digits used when the first 3002 positive even integers are written? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11456",
"style": "rule"
} | {
"index": "37264",
"split": "train"
} |
math | [
{
"content": "Three circles are drawn around vertices \\( A, B, \\) and \\( C \\) of a regular hexagon \\( ABCDEF \\) with side length 2 units, such that the circles touch each other externally. What is the radius of the smallest circle? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "2 - \\sqrt{3}",
"style": "rule"
} | {
"index": "37265",
"split": "train"
} |
math | [
{
"content": "On the board are written 5 integers. By adding them in pairs, the following set of 10 numbers was obtained: $5, 8, 9, 13, 14, 14, 15, 17, 18, 23$. Determine which numbers are written on the board. In the answer, write their product. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "4752",
"style": "rule"
} | {
"index": "37266",
"split": "train"
} |
math | [
{
"content": "Given that the sequence $\\{a\\_n\\}$ is a monotonically increasing arithmetic sequence, the probability that any three terms taken from $\\{a\\_1, a\\_2, a\\_3, a\\_4, a\\_5, a\\_6, a\\_7\\}$ will leave the remaining four terms as a monotonically increasing arithmetic sequence is $\\_\\_\\_\\_\\_... | {
"ground_truth": "\\frac{1}{7}",
"style": "rule"
} | {
"index": "37267",
"split": "train"
} |
math | [
{
"content": "Find all positive integers $a$ such that there exists a set $X$ of $6$ integers satisfying the following conditions: for every $k=1,2,\\ldots ,36$ there exist $x,y\\in X$ such that $ax+y-k$ is divisible by $37$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6, 31",
"style": "rule"
} | {
"index": "37268",
"split": "train"
} |
math | [
{
"content": "Two Wei Qi teams, $A$ and $B$, each comprising 7 members, compete against each other. Players from each team face off in sequence. The first game is between the first player of each team. The loser is eliminated, and the winner moves on to face the next player of the opposing team. This process co... | {
"ground_truth": "3432",
"style": "rule"
} | {
"index": "37269",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\vec{a}$ and $\\vec{b}$ with magnitudes $|\\vec{a}|=2$ and $|\\vec{b}|=\\sqrt{3}$, respectively, and the equation $( \\vec{a}+2\\vec{b}) \\cdot ( \\vec{b}-3\\vec{a})=9$:\n\n(1) Find the dot product $\\vec{a} \\cdot \\vec{b}$.\n\n(2) In triangle $ABC$, with $\\vec{AB}=\\vec{a}$ and ... | {
"ground_truth": "-\\sqrt{3}",
"style": "rule"
} | {
"index": "37270",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of all positive integer divisors of $129,600$. Calculate the number of numbers that are the product of two distinct elements of $S$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "488",
"style": "rule"
} | {
"index": "37271",
"split": "train"
} |
math | [
{
"content": "A person rolls two dice simultaneously and gets the scores $a$ and $b$. The eccentricity $e$ of the ellipse $\\frac{y^2}{a^2} + \\frac{x^2}{b^2} = 1$ satisfies $e \\geq \\frac{\\sqrt{3}}{2}$. Calculate the probability that this event occurs. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "37272",
"split": "train"
} |
math | [
{
"content": "The number \\( 20! = 1 \\cdot 2 \\cdot \\ldots \\cdot 20 = 2432902008176640000 \\) has 41,040 natural divisors. How many of them are odd? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2160",
"style": "rule"
} | {
"index": "37273",
"split": "train"
} |
math | [
{
"content": "A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\\frac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integer... | {
"ground_truth": "0.4",
"style": "rule"
} | {
"index": "37274",
"split": "train"
} |
math | [
{
"content": "Find the quadratic polynomial, with real coefficients, which has $-2 - 3i$ as a root, and where the coefficient of $x$ is $-4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-x^2 - 4x - 13",
"style": "rule"
} | {
"index": "37275",
"split": "train"
} |
math | [
{
"content": "Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives\n\\[{1000 \\choose 0}(0.2)^0+{1000 \\choose 1}(0.2)^1+{1000 \\choose 2}(0.2)^2+\\cdots+{1000 \\choose 1000}(0.2)^{1000}= A_0 + A_1 + A_2 + \\cdots + A_{1000},\\]where $A_k = {1000 \\choose k}(0.2)^k$ for $k ... | {
"ground_truth": "166",
"style": "rule"
} | {
"index": "37276",
"split": "train"
} |
math | [
{
"content": "Given that f(x) = |log₃x|, if a and b satisfy f(a - 1) = f(2b - 1), and a ≠ 2b, then the minimum value of a + b is ___. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2} + \\sqrt{2}",
"style": "rule"
} | {
"index": "37277",
"split": "train"
} |
math | [
{
"content": "There are 8 different positive integers. Among them, at least 6 are multiples of 2, at least 5 are multiples of 3, at least 3 are multiples of 5, and at least 1 is a multiple of 7. To minimize the largest number among these 8 integers, what is this largest number? Please output the final answer wi... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "37278",
"split": "train"
} |
math | [
{
"content": "Let $G$ be the centroid of triangle $ABC,$ and let $P$ be an arbitrary point. Then there exists a constant $k$ so that\n\\[PA^2 + PB^2 + PC^2 = k \\cdot PG^2 + GA^2 + GB^2 + GC^2.\\]Find $k.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "37279",
"split": "train"
} |
math | [
{
"content": "In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. Given the equation acosB - bcosA + 2c = 0, find the value of $\\frac{tanA}{tanB}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{3}",
"style": "rule"
} | {
"index": "37280",
"split": "train"
} |
math | [
{
"content": "In a village where Glafira lives, there is a small pond that is filled by springs at the bottom. Curious Glafira found out that a herd of 17 cows completely drank this pond in 3 days. After some time, the springs filled the pond again, and then 2 cows drank it in 30 days. How many days will it tak... | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "37281",
"split": "train"
} |
math | [
{
"content": "Let $P$ be the parabola with equation $y=x^2$ and let $Q = (20, 14)$. There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s.$ What is $r + s$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "37282",
"split": "train"
} |
math | [
{
"content": "The carbon dioxide emissions in a certain region reach a peak of a billion tons (a > 0) and then begin to decline. The relationship between the carbon dioxide emissions S (in billion tons) and time t (in years) satisfies the function S = a · b^t. If after 7 years, the carbon dioxide emissions are ... | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "37283",
"split": "train"
} |
math | [
{
"content": "Given the line $y=kx+b\\left(k \\gt 0\\right)$ is tangent to the circle $x^{2}+y^{2}=1$ and the circle $\\left(x-4\\right)^{2}+y^{2}=1$, find $k=$____ and $b=$____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "37284",
"split": "train"
} |
math | [
{
"content": "There are $7$ students, among whom $A_{1}$, $A_{2}$, $A_{3}$ excel in mathematics, $B_{1}$, $B_{2}$ excel in physics, and $C_{1}$, $C_{2}$ excel in chemistry. One student with excellent performance in mathematics, one in physics, and one in chemistry will be selected to form a team to represent th... | {
"ground_truth": "\\frac{5}{6}",
"style": "rule"
} | {
"index": "37285",
"split": "train"
} |
math | [
{
"content": "Find an eight-digit palindrome that is a multiple of three, composed of the digits 0 and 1, given that all its prime divisors only use the digits 1, 3, and %. (Palindromes read the same forwards and backwards, for example, 11011). Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "10111101",
"style": "rule"
} | {
"index": "37286",
"split": "train"
} |
math | [
{
"content": "Solve for $r$ in the equation $19-3=2+r$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "37287",
"split": "train"
} |
math | [
{
"content": "A positive number is increased by $60\\%$. By what percentage should the result be decreased to return to the original value? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37.5\\%",
"style": "rule"
} | {
"index": "37288",
"split": "train"
} |
math | [
{
"content": "The base of a triangle is $80$ , and one side of the base angle is $60^\\circ$ . The sum of the lengths of the other two sides is $90$ . The shortest side is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "37289",
"split": "train"
} |
math | [
{
"content": "Positive real numbers \\( a, b, c \\) are in a geometric progression \\((q \\neq 1)\\), and \\( \\log _{a} b, \\log _{b} c, \\log _{c} a \\) are in an arithmetic progression. Find the common difference \\( d \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{2}",
"style": "rule"
} | {
"index": "37290",
"split": "train"
} |
math | [
{
"content": "Somewhere in the universe, $n$ students are taking a 10-question math competition. Their collective performance is called laughable if, for some pair of questions, there exist 57 students such that either all of them answered both questions correctly or none of them answered both questions correct... | {
"ground_truth": "253",
"style": "rule"
} | {
"index": "37291",
"split": "train"
} |
math | [
{
"content": "A quadrilateral has vertices $P(a,b)$, $Q(2b,a)$, $R(-a, -b)$, and $S(-2b, -a)$, where $a$ and $b$ are integers and $a>b>0$. Find the area of quadrilateral $PQRS$.\n**A)** $16$ square units \n**B)** $20$ square units \n**C)** $24$ square units \n**D)** $28$ square units \n**E)** $32$ square units ... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "37292",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of\n\\[\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "37293",
"split": "train"
} |
math | [
{
"content": "Simplify $(2^8 + 4^5)(2^3 - (-2)^2)^{11}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5368709120",
"style": "rule"
} | {
"index": "37294",
"split": "train"
} |
math | [
{
"content": "Find the number of permutations of the letters $ABCDE$ where the letters $A$ and $B$ are not adjacent and the letters $C$ and $D$ are not adjacent. For example, count the permutations $ACBDE$ and $DEBCA$ but not $ABCED$ or $EDCBA$ . Please output the final answer within \\boxed{}.... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "37295",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle and $k$ be a positive number such that altitudes $AD$, $BE$, and $CF$ are extended past $A$, $B$, and $C$ to points $A'$, $B'$, and $C'$ respectively, where $AA' = kBC$, $BB' = kAC$, and $CC' = kAB$. Suppose further that $A''$ is a point such that the line segment $AA''$ is... | {
"ground_truth": "\\frac{1}{\\sqrt{3}}",
"style": "rule"
} | {
"index": "37296",
"split": "train"
} |
math | [
{
"content": "Find all prime numbers $p$ such that $y^{2}=x^{3}+4x$ has exactly $p$ solutions in integers modulo $p$. In other words, determine all prime numbers $p$ with the following property: there exist exactly $p$ ordered pairs of integers $(x, y)$ such that $x, y \\in\\{0,1, \\ldots, p-1\\}$ and $p \\text... | {
"ground_truth": "p=2 \\text{ and } p \\equiv 3(\\bmod 4)",
"style": "rule"
} | {
"index": "37297",
"split": "train"
} |
math | [
{
"content": "Note that $9^2 = 81$, which contains no zeros; $99^2 = 9801$, which contains 1 zero; and $999^2 = 998,\\!001$, which contains 2 zeros. Assuming this pattern continues, how many zeros are in the expansion of $99,\\!999,\\!999^2$? Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "37298",
"split": "train"
} |
math | [
{
"content": "Santa Claus arrived at the house of Arnaldo and Bernaldo carrying ten distinct toys numbered from 1 to 10 and said to them: \"Toy number 1 is for you, Arnaldo, and toy number 2 is for you, Bernaldo. But this year, you may choose to keep more toys as long as you leave at least one for me.\" Determi... | {
"ground_truth": "6305",
"style": "rule"
} | {
"index": "37299",
"split": "train"
} |
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