data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given that $\\cos ( \\frac {π}{6}+α) \\cdot \\cos ( \\frac {π}{3}-α)=- \\frac {1}{4}$, where $α \\in ( \\frac {π}{3}, \\frac {π}{2})$, find the value of $\\sin 2α$ and the value of $\\tan α - \\frac {1}{\\tan α}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "35500",
"split": "train"
} |
math | [
{
"content": "A gymnastics team consists of 48 members. To form a square formation, they need to add at least ____ people or remove at least ____ people. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35501",
"split": "train"
} |
math | [
{
"content": "Find the value of $x$ between 0 and 180 such that\n\\[\\tan (120^\\circ - x^\\circ) = \\frac{\\sin 120^\\circ - \\sin x^\\circ}{\\cos 120^\\circ - \\cos x^\\circ}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "35502",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=81$ and $a_n= \\begin{cases} -1+\\log_{3}a_{n-1}, & n=2k \\\\ 3^{a_{n-1}}, & n=2k+1 \\end{cases}$ (where $k\\in\\mathbb{N}^*$), find the maximum value of the sum of the first $n$ terms of the sequence, $S_n$. Please output the final answer within \\boxe... | {
"ground_truth": "127",
"style": "rule"
} | {
"index": "35503",
"split": "train"
} |
math | [
{
"content": "A sequence \\(a_1\\), \\(a_2\\), \\(\\ldots\\) of non-negative integers is defined by the rule \\(a_{n+2}=|a_{n+1}-a_n|\\) for \\(n\\geq1\\). If \\(a_1=1010\\), \\(a_2<1010\\), and \\(a_{2023}=0\\), how many different values of \\(a_2\\) are possible? Please output the final answer within \\boxed{... | {
"ground_truth": "399",
"style": "rule"
} | {
"index": "35504",
"split": "train"
} |
math | [
{
"content": "In a bag containing 12 green marbles and 8 purple marbles, Phil draws a marble at random, records its color, replaces it, and repeats this process until he has drawn 10 marbles. What is the probability that exactly five of the marbles he draws are green? Express your answer as a decimal rounded to... | {
"ground_truth": "0.201",
"style": "rule"
} | {
"index": "35505",
"split": "train"
} |
math | [
{
"content": "On each side of a square, a point is taken. It turns out that these points are the vertices of a rectangle whose sides are parallel to the diagonals of the square. Find the perimeter of the rectangle if the diagonal of the square is 6. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35506",
"split": "train"
} |
math | [
{
"content": "It is known that ship $A$ is located at $80^{\\circ}$ north by east from lighthouse $C$, and the distance from $A$ to $C$ is $2km$. Ship $B$ is located at $40^{\\circ}$ north by west from lighthouse $C$, and the distance between ships $A$ and $B$ is $3km$. Find the distance from $B$ to $C$ in $km$... | {
"ground_truth": "\\sqrt {6}-1",
"style": "rule"
} | {
"index": "35507",
"split": "train"
} |
math | [
{
"content": "Phoenix hiked the Rocky Path Trail last week. It took four days to complete the trip. The first two days she hiked a total of 22 miles. The second and third days she averaged 13 miles per day. The last two days she hiked a total of 30 miles. The total hike for the first and third days was 26 miles... | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "35508",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered triples of divisors $(d_{1}, d_{2}, d_{3})$ of 360 such that $d_{1} d_{2} d_{3}$ is also a divisor of 360. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "800",
"style": "rule"
} | {
"index": "35509",
"split": "train"
} |
math | [
{
"content": "Find the set of all attainable values of $\\frac{ab+b^{2}}{a^{2}+b^{2}}$ for positive real $a, b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left(0, \\frac{1+\\sqrt{2}}{2}\\right]",
"style": "rule"
} | {
"index": "35510",
"split": "train"
} |
math | [
{
"content": "Two numbers in the $4 \\times 4$ grid can be swapped to create a Magic Square (in which all rows, all columns and both main diagonals add to the same total).\nWhat is the sum of these two numbers?\nA 12\nB 15\nC 22\nD 26\nE 28\n\n\\begin{tabular}{|c|c|c|c|}\n\\hline \n9 & 6 & 3 & 16 \\\\\n\\hline ... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "35511",
"split": "train"
} |
math | [
{
"content": "Calculate the circulation of the vector field:\n\na) $\\vec{A}=x^{2} y^{2} \\vec{i}+\\vec{j}+z \\vec{k}$ along the circle $x^{2}+y^{2}=a^{2}, z=0$;\n\nb) $\\dot{A}=(x-2 z) \\dot{i}+(x+3 y+z) \\dot{j}+(5 x+y) \\vec{k}$ along the perimeter of the triangle $A B C$ with vertices $A(1,0,0), B(0,1,0), C... | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "35512",
"split": "train"
} |
math | [
{
"content": "Find the total number of sets of positive integers \\((x, y, z)\\), where \\(x, y\\) and \\(z\\) are positive integers, with \\(x < y < z\\) such that\n$$\nx + y + z = 203.\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3333",
"style": "rule"
} | {
"index": "35513",
"split": "train"
} |
math | [
{
"content": "For which values of the parameter \\(a\\) does the equation \\(x^{3} + 16x^{2} + ax + 64 = 0\\) have three distinct real roots that form a geometric progression? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "35514",
"split": "train"
} |
math | [
{
"content": "Compute the limit of the function:\n\n$$\n\\lim _{x \\rightarrow 0} \\frac{4^{5 x}-9^{-2 x}}{\\sin x-\\operatorname{tg}(x^{3})}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\ln (1024 \\cdot 81)",
"style": "rule"
} | {
"index": "35515",
"split": "train"
} |
math | [
{
"content": "Define $f\\left(n\\right)=\\textrm{LCM}\\left(1,2,\\ldots,n\\right)$ . Determine the smallest positive integer $a$ such that $f\\left(a\\right)=f\\left(a+2\\right)$ .\n\n*2017 CCA Math Bonanza Lightning Round #2.4* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "35516",
"split": "train"
} |
math | [
{
"content": "Let $P_0(x) = x^3 + 313x^2 - 77x - 8\\,$. For integers $n \\ge 1\\,$, define $P_n(x) = P_{n - 1}(x - n)\\,$. What is the coefficient of $x\\,$ in $P_{20}(x)\\,$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "763",
"style": "rule"
} | {
"index": "35517",
"split": "train"
} |
math | [
{
"content": "A wooden cube has edges of length $3$ meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is Please output the final answer... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "35518",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $2\\sin 2A+\\sin (A-B)=\\sin C$, and $A\\neq \\frac{\\pi}{2}$.\n- (I) Find the value of $\\frac{a}{b}$;\n- (II) If $c=2$ and $C= \\frac{\\pi}{3}$, find the area of $\\triangle A... | {
"ground_truth": "\\frac{2 \\sqrt {3}}{3}",
"style": "rule"
} | {
"index": "35519",
"split": "train"
} |
math | [
{
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 1, form a dihedral angle of 45 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing this ed... | {
"ground_truth": "\\frac{\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "35520",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "248",
"style": "rule"
} | {
"index": "35521",
"split": "train"
} |
math | [
{
"content": "Elective 4-5: Selected Topics on Inequalities. Given the function $f(x) = |2x-1| + |2x+3|$.\n\n$(1)$ Solve the inequality $f(x) \\geqslant 6$;\n\n$(2)$ Let the minimum value of $f(x)$ be $m$, and let the positive real numbers $a, b$ satisfy $2ab + a + 2b = m$. Find the minimum value of $a + 2b$. P... | {
"ground_truth": "2\\sqrt{5} - 2",
"style": "rule"
} | {
"index": "35522",
"split": "train"
} |
math | [
{
"content": "Any six points are taken inside or on an equilateral triangle with side length 1. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Then $b$ is: Please o... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "35523",
"split": "train"
} |
math | [
{
"content": "Among 50 school teams participating in the HKMO, no team answered all four questions correctly. The first question was solved by 45 teams, the second by 40 teams, the third by 35 teams, and the fourth by 30 teams. How many teams solved both the third and the fourth questions? Please output the fin... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "35524",
"split": "train"
} |
math | [
{
"content": "The plane angle at the vertex of a regular triangular pyramid is $90^{\\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "35525",
"split": "train"
} |
math | [
{
"content": "On graph paper, a stepwise right triangle was drawn with legs equal to 6 cells each. Then, all grid lines inside the triangle were outlined. What is the maximum number of rectangles that can be found in this drawing? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "126",
"style": "rule"
} | {
"index": "35526",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c \\sin A = \\sqrt{3}a \\cos C$ and $(a-c)(a+c)=b(b-c)$, find the period and the monotonically increasing interval of the function $f(x) = 2 \\sin x \\cos (\\frac{\\pi}{2} - x) - \\s... | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "35527",
"split": "train"
} |
math | [
{
"content": "Given 6 teachers who will be allocated to two classes, where the maximum number of teachers in each class is 4, determine the number of different arrangements. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "35528",
"split": "train"
} |
math | [
{
"content": "In a debate competition with four students participating, the rules are as follows: Each student must choose one question to answer from two given topics, Topic A and Topic B. For Topic A, answering correctly yields 100 points and answering incorrectly results in a loss of 100 points. For Topic B,... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "35529",
"split": "train"
} |
math | [
{
"content": "First, factorize 42 and 30 into prime factors, then answer the following questions: \n(1) 42= , 30= . \n(2) The common prime factors of 42 and 30 are . \n(3) The unique prime factors of 42 and 30 are . \n(4) The greatest common divisor (GCD) of 42 and 30 is . \n(5) The least... | {
"ground_truth": "210",
"style": "rule"
} | {
"index": "35530",
"split": "train"
} |
math | [
{
"content": "Person A and person B each have a certain number of books. If person A gives 10 books to person B, then the total number of books between the two of them will be equal. If person B gives 10 books to person A, then the number of books person A has will be twice the number of books person B has left... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "35531",
"split": "train"
} |
math | [
{
"content": "Given that the solution set for the inequality $ax^2+ax+2>0$ is $\\mathbb{R}$ (the set of all real numbers), let the set of all numerical values of the real number $a$ be denoted as $M$.\n(1) Find the set $M$.\n(2) If $t>0$, for all $a \\in M$, it holds that $(a^2-2a)t \\leq t^2 + 3t - 46$. Find t... | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "35532",
"split": "train"
} |
math | [
{
"content": "The midpoints of the sides of a triangle with area $T$ are joined to form a triangle with area $M$. What is the ratio of $M$ to $T$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "35533",
"split": "train"
} |
math | [
{
"content": "For positive real numbers $s$, let $\\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\\tau(s)$ is nonempty, but all triangles in $\\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$. Please output... | {
"ground_truth": "736",
"style": "rule"
} | {
"index": "35534",
"split": "train"
} |
math | [
{
"content": "For natural numbers $m$ greater than or equal to 2, the decomposition of their cube powers can be represented as follows: $2^3 = 3 + 5$, $3^3 = 7 + 9 + 11$, $4^3 = 13 + 15 + 17 + 19$. Then, \n(1) The smallest number in the decomposition of $8^3$ is; \n(2) Following the above pattern, the $n$-th ... | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "35535",
"split": "train"
} |
math | [
{
"content": "A student, Leo, needs to earn 30 study points for a special credit. For the first 6 points, he needs to complete 1 project each. For the next 6 points, he needs 2 projects each; for the next 6 points, 3 projects each, and so on. Determine the minimum number of projects Leo needs to complete to ear... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "35536",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$ points $D$ and $E$ lie on $\\overline{BC}$ and $\\overline{AC}$, respectively. If $\\overline{AD}$ and $\\overline{BE}$ intersect at $T$ so that $AT/DT=3$ and $BT/ET=4$, what is $CD/BD$?\n\n[asy]\npair A,B,C,D,I,T;\nA=(0,0);\nB=(6,8);\nC=(11,0);\nD=(9.33,2.66);\nI=(7.5,0);\nT=... | {
"ground_truth": "\\frac{4}{11}",
"style": "rule"
} | {
"index": "35537",
"split": "train"
} |
math | [
{
"content": "A triangle $H$ is inscribed in a regular hexagon $S$ such that one side of $H$ is parallel to one side of $S$. What is the maximum possible ratio of the area of $H$ to the area of $S$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3/8",
"style": "rule"
} | {
"index": "35538",
"split": "train"
} |
math | [
{
"content": "John learned that Lisa scored exactly 85 on the American High School Mathematics Examination (AHSME). Due to this information, John was able to determine exactly how many problems Lisa solved correctly. If Lisa's score had been any lower but still over 85, John would not have been able to determin... | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "35539",
"split": "train"
} |
math | [
{
"content": "$\\frac{\\text{华杯赛}}{\\text{少} \\times \\text{俊} + \\text{金坛} + \\text{论} \\times \\text{数}} = 15$\n\nIn the above equation, different Chinese characters represent different digits between $1$ and $9$. When the three-digit number \"华杯赛\" reaches its maximum value, please write a solution where the... | {
"ground_truth": "975",
"style": "rule"
} | {
"index": "35540",
"split": "train"
} |
math | [
{
"content": "Ainsley and Buddy play a game where they repeatedly roll a standard fair six-sided die. Ainsley wins if two multiples of 3 in a row are rolled before a non-multiple of 3 followed by a multiple of 3, and Buddy wins otherwise. If the probability that Ainsley wins is $\\frac{a}{b}$ for relatively pri... | {
"ground_truth": "109",
"style": "rule"
} | {
"index": "35541",
"split": "train"
} |
math | [
{
"content": "Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "236",
"style": "rule"
} | {
"index": "35542",
"split": "train"
} |
math | [
{
"content": "In the geometric sequence with a first term of $6$ and a second term of $-6$, what is the $205^{th}$ term? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "35543",
"split": "train"
} |
math | [
{
"content": "Given a circle $O: x^2 + y^2 = 6$, and $P$ is a moving point on circle $O$. A perpendicular line $PM$ is drawn from $P$ to the x-axis at $M$, and $N$ is a point on $PM$ such that $\\overrightarrow{PM} = \\sqrt{2} \\overrightarrow{NM}$.\n\n(Ⅰ) Find the equation of the trajectory $C$ of point $N$; ... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "35544",
"split": "train"
} |
math | [
{
"content": "A frog starts climbing from the bottom of a 12-meter deep well at 8:00 AM. For every 3 meters it climbs up, it slides down 1 meter due to the slippery walls. The time to slide down 1 meter is one-third the time taken to climb up 3 meters. At 8:17 AM, the frog reaches 3 meters from the well's top f... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "35545",
"split": "train"
} |
math | [
{
"content": "The reciprocal of $\\frac{2}{3}$ is ______, the opposite of $-2.5$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.5",
"style": "rule"
} | {
"index": "35546",
"split": "train"
} |
math | [
{
"content": "How many distinct arrangements of the letters in the word \"balloon\" are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1260",
"style": "rule"
} | {
"index": "35547",
"split": "train"
} |
math | [
{
"content": "Given that the function $f(x)=\\sin (ωx+φ)(ω > 0,0 < φ < π)$ has a distance of $\\frac {π}{2}$ between adjacent symmetry axes, and the function $y=f(x+ \\frac {π}{2})$ is an even function.\n1. Find the analytical expression of $f(x)$.\n2. If $α$ is an acute angle, and $f(\\frac {α}{2}+ \\frac {π}{... | {
"ground_truth": "\\frac {24+7 \\sqrt {3}}{50}",
"style": "rule"
} | {
"index": "35548",
"split": "train"
} |
math | [
{
"content": "How many even integers between 3000 and 6000 have four different digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "784",
"style": "rule"
} | {
"index": "35549",
"split": "train"
} |
math | [
{
"content": "Given a positive number \\(r\\) such that the set \\(T=\\left\\{(x, y) \\mid x, y \\in \\mathbf{R}\\right.\\) and \\(\\left.x^{2}+(y-7)^{2} \\leqslant r^{2}\\right\\}\\) is a subset of the set \\(S=\\{(x, y) \\mid x, y \\in \\mathbf{R}\\right.\\) and for any \\(\\theta \\in \\mathbf{R}\\), \\(\\co... | {
"ground_truth": "4 \\sqrt{2}",
"style": "rule"
} | {
"index": "35550",
"split": "train"
} |
math | [
{
"content": "Let \\( a \\) and \\( b \\) be positive real numbers. Given that \\(\\frac{1}{a} + \\frac{1}{b} \\leq 2\\sqrt{2}\\) and \\((a - b)^2 = 4(ab)^3\\), find \\(\\log_a b\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "35551",
"split": "train"
} |
math | [
{
"content": "Find the greatest common divisor of 75 and 360. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "35552",
"split": "train"
} |
math | [
{
"content": "In the diagram, six squares form a \\( 2 \\times 3 \\) grid. The middle square in the top row is marked with an \\( R \\). Each of the five remaining squares is to be marked with an \\( R \\), \\( S \\), or \\( T \\). In how many ways can the grid be completed so that it includes at least one pair... | {
"ground_truth": "225",
"style": "rule"
} | {
"index": "35553",
"split": "train"
} |
math | [
{
"content": "Side $\\overline{A B}$ of $\\triangle A B C$ is the diameter of a semicircle, as shown below. If $A B=3+\\sqrt{3}, B C=3 \\sqrt{2}$, and $A C=2 \\sqrt{3}$, then the area of the shaded region can be written as $\\frac{a+(b+c \\sqrt{d}) \\pi}{e}$, where $a, b, c, d, e$ are integers, $e$ is positive,... | {
"ground_truth": "147938",
"style": "rule"
} | {
"index": "35554",
"split": "train"
} |
math | [
{
"content": "In equilateral $\\triangle ABC$ let points $D$ and $E$ trisect $\\overline{BC}$. Then $\\sin(\\angle DAE)$ can be expressed in the form $\\frac{a\\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "35555",
"split": "train"
} |
math | [
{
"content": "For the infinite series $1-\\frac12-\\frac14+\\frac18-\\frac{1}{16}-\\frac{1}{32}+\\frac{1}{64}-\\frac{1}{128}-\\cdots$ let $S$ be the (limiting) sum. Then $S$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{7}",
"style": "rule"
} | {
"index": "35556",
"split": "train"
} |
math | [
{
"content": "In a bookshelf, there are four volumes of Astrid Lindgren's collected works in order, each containing 200 pages. A little worm living in these volumes burrowed a path from the first page of the first volume to the last page of the fourth volume. How many pages did the worm burrow through? Please o... | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "35557",
"split": "train"
} |
math | [
{
"content": "What is the least value of $y$ such that $3y^2 + 5y + 2 = 4$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "35558",
"split": "train"
} |
math | [
{
"content": "Find 100 times the area of a regular dodecagon inscribed in a unit circle. Round your answer to the nearest integer if necessary.\n[asy]\ndefaultpen(linewidth(0.7)); real theta = 17; pen dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);\ndraw(unitcircle,dg);\nfor(int i = 0; i <... | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "35559",
"split": "train"
} |
math | [
{
"content": "For how many positive integers $n$ less than or equal to 500 is $$(\\cos t - i\\sin t)^n = \\cos nt - i\\sin nt$$ true for all real $t$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "35560",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\log_5625$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "35561",
"split": "train"
} |
math | [
{
"content": "There is a settlement $C$ (a point) located at the intersection of roads $A$ and $B$ (straight lines). Sasha walks along road $A$ towards $C$, taking 45 steps per minute with a step length of 60 cm. At the start, Sasha is 290 m away from $C$. Dania walks along road $B$ towards $C$ at a rate of 55 ... | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "35562",
"split": "train"
} |
math | [
{
"content": "Let $p_{i}$ be the $i$th prime. Let $$f(x)=\\sum_{i=1}^{50} p_{i} x^{i-1}=2+3x+\\cdots+229x^{49}$$ If $a$ is the unique positive real number with $f(a)=100$, estimate $A=\\lfloor 100000a\\rfloor$. An estimate of $E$ will earn $\\max (0,\\lfloor 20-|A-E| / 250\\rfloor)$ points. Please output the fi... | {
"ground_truth": "83601",
"style": "rule"
} | {
"index": "35563",
"split": "train"
} |
math | [
{
"content": "A four-digit natural number $M$, where the digits in each place are not $0$, we take its hundreds digit as the tens digit and the tens digit as the units digit to form a new two-digit number. If this two-digit number is greater than the sum of the thousands digit and units digit of $M$, then we ca... | {
"ground_truth": "5883",
"style": "rule"
} | {
"index": "35564",
"split": "train"
} |
math | [
{
"content": "Let $ a, b, c, d,m, n \\in \\mathbb{Z}^\\plus{}$ such that \\[ a^2\\plus{}b^2\\plus{}c^2\\plus{}d^2 \\equal{} 1989,\\]\n\\[ a\\plus{}b\\plus{}c\\plus{}d \\equal{} m^2,\\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$ Please output the final answer ... | {
"ground_truth": "m = 9,n = 6",
"style": "rule"
} | {
"index": "35565",
"split": "train"
} |
math | [
{
"content": "Consider the graph of \\( y = g(x) \\), with \\( 1 \\) unit between grid lines, where \\( g(x) = \\frac{(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{720} - 2.5 \\), defined only on the shown domain. \n\nDetermine the sum of all integers \\( c \\) for which the equation \\( g(x) = c \\) has exactly \\( 4 \\) solu... | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "35566",
"split": "train"
} |
math | [
{
"content": "How many 4-letter words with at least one consonant can be constructed from the letters $A$, $B$, $C$, $D$, and $E$? (Note that $B$, $C$, and $D$ are consonants, any word is valid, not just English language words, and letters may be used more than once.) Please output the final answer within \\bo... | {
"ground_truth": "609",
"style": "rule"
} | {
"index": "35567",
"split": "train"
} |
math | [
{
"content": "A circle is inscribed in a convex quadrilateral \\(ABCD\\) with its center at point \\(O\\), and \\(AO=OC\\). Additionally, \\(BC=5\\), \\(CD=12\\), and \\(\\angle DAB\\) is a right angle.\n\nFind the area of the quadrilateral \\(ABCD\\). Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "35568",
"split": "train"
} |
math | [
{
"content": "Seven identical bowling balls weigh the same as three identical canoes. If one of the canoes weighs a total of 28 pounds, how many pounds does one of the bowling balls weigh? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35569",
"split": "train"
} |
math | [
{
"content": "Arrange the digits \\(1, 2, 3, 4, 5, 6, 7, 8, 9\\) in some order to form a nine-digit number \\(\\overline{\\text{abcdefghi}}\\). If \\(A = \\overline{\\text{abc}} + \\overline{\\text{bcd}} + \\overline{\\text{cde}} + \\overline{\\text{def}} + \\overline{\\text{efg}} + \\overline{\\text{fgh}} + \\... | {
"ground_truth": "4648",
"style": "rule"
} | {
"index": "35570",
"split": "train"
} |
math | [
{
"content": "There are positive integers $x$ and $y$ that satisfy the system of equations \\begin{align*} \\log_{10} x + 2 \\log_{10} (\\text{gcd}(x,y)) &= 60\\\\ \\log_{10} y + 2 \\log_{10} (\\text{lcm}(x,y)) &= 570. \\end{align*} Let $m$ be the number of (not necessarily distinct) prime factors in the prime ... | {
"ground_truth": "880",
"style": "rule"
} | {
"index": "35571",
"split": "train"
} |
math | [
{
"content": "Circles $\\omega_1$ and $\\omega_2$ intersect at points $X$ and $Y$. Line $\\ell$ is tangent to $\\omega_1$ and $\\omega_2$ at $A$ and $B$, respectively, with line $AB$ closer to point $X$ than to $Y$. Circle $\\omega$ passes through $A$ and $B$ intersecting $\\omega_1$ again at $D \\neq A$ and in... | {
"ground_truth": "270",
"style": "rule"
} | {
"index": "35572",
"split": "train"
} |
math | [
{
"content": "At a community gathering there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\\frac{3}{7}$. Calculate the fraction of the people in the gathering who are married men. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "\\frac{4}{11}",
"style": "rule"
} | {
"index": "35573",
"split": "train"
} |
math | [
{
"content": "Mike had a bag of candies, and all candies were whole pieces that cannot be divided. Initially, Mike ate $\\frac{1}{4}$ of the candies. Then, he shared $\\frac{1}{3}$ of the remaining candies with his sister, Linda. Next, both Mike and his father ate 12 candies each from the remaining candies Mike... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "35574",
"split": "train"
} |
math | [
{
"content": "In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is[i] color-identifiable[/i] if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$.\n\nFor all positive integ... | {
"ground_truth": "\\lceil \\frac{n}{t} \\rceil",
"style": "rule"
} | {
"index": "35575",
"split": "train"
} |
math | [
{
"content": "The centers of the three circles A, B, and C are collinear with the center of circle B lying between the centers of circles A and C. Circles A and C are both externally tangent to circle B, and the three circles share a common tangent line. Given that circle A has radius $12$ and circle B has ra... | {
"ground_truth": "147",
"style": "rule"
} | {
"index": "35576",
"split": "train"
} |
math | [
{
"content": "At the round table, $10$ people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) . It is clear thath I have at least one knight and at least one liar. \nWhat is the largest number of those sitting at the table can say: ''Both of my nei... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "35577",
"split": "train"
} |
math | [
{
"content": "Express $249_{11}+3AB_{12}$ as a base 10 integer. Here $A$ and $B$ denote the digits whose values are 10 and 11, respectively, in base 12. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "858",
"style": "rule"
} | {
"index": "35578",
"split": "train"
} |
math | [
{
"content": "Given that the function $f(x)$ satisfies $f(x+y)=f(x)+f(y)$ for all real numbers $x, y \\in \\mathbb{R}$, and $f(x) < 0$ when $x > 0$, and $f(3)=-2$.\n1. Determine the parity (odd or even) of the function.\n2. Determine the monotonicity of the function on $\\mathbb{R}$.\n3. Find the maximum and mi... | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "35579",
"split": "train"
} |
math | [
{
"content": "One of the roots of\n\\[ax^3 + 3x^2 + bx - 65 = 0,\\]is $-2 - 3i,$ where $a$ and $b$ are real numbers. Find the real root of this cubic polynomial. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "35580",
"split": "train"
} |
math | [
{
"content": "Find the sum of the digits of the number \\( A \\), if \\( A=2^{63} \\cdot 4^{25} \\cdot 5^{106}-2^{22} \\cdot 4^{44} \\cdot 5^{105}-1 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "959",
"style": "rule"
} | {
"index": "35581",
"split": "train"
} |
math | [
{
"content": "The exact amount of fencing that enclosed the four congruent equilateral triangular corrals shown here is reused to form one large equilateral triangular corral. What is the ratio of the total area of the four small corrals to the area of the new large corral? Express your answer as a common fract... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "35582",
"split": "train"
} |
math | [
{
"content": "Riley has 64 cubes with dimensions \\(1 \\times 1 \\times 1\\). Each cube has its six faces labeled with a 2 on two opposite faces and a 1 on each of its other four faces. The 64 cubes are arranged to build a \\(4 \\times 4 \\times 4\\) cube. Riley determines the total of the numbers on the outsid... | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "35583",
"split": "train"
} |
math | [
{
"content": "What is the value of $1234 + 2341 + 3412 + 4123$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11110",
"style": "rule"
} | {
"index": "35584",
"split": "train"
} |
math | [
{
"content": "Li Qiang rented a piece of land from Uncle Zhang, for which he has to pay Uncle Zhang 800 yuan and a certain amount of wheat every year. One day, he did some calculations: at that time, the price of wheat was 1.2 yuan per kilogram, which amounted to 70 yuan per mu of land; but now the price of whe... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "35585",
"split": "train"
} |
math | [
{
"content": "Given that the central angle of a sector is $\\frac{3}{2}$ radians, and its radius is 6 cm, then the arc length of the sector is \\_\\_\\_\\_\\_\\_ cm, and the area of the sector is \\_\\_\\_\\_\\_\\_ cm<sup>2</sup>. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "35586",
"split": "train"
} |
math | [
{
"content": "John wants to find all the five-letter words that begin and end with the same letter. How many combinations of letters satisfy this property? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "456976",
"style": "rule"
} | {
"index": "35587",
"split": "train"
} |
math | [
{
"content": "An integer has exactly 4 prime factors, and the sum of the squares of these factors is 476. Find this integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1989",
"style": "rule"
} | {
"index": "35588",
"split": "train"
} |
math | [
{
"content": "A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$ Please output th... | {
"ground_truth": "77",
"style": "rule"
} | {
"index": "35589",
"split": "train"
} |
math | [
{
"content": "Find all prime numbers $p,q,r$ , such that $\\frac{p}{q}-\\frac{4}{r+1}=1$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\[\n(7, 3, 2), (3, 2, 7), (5, 3, 5)\n\\]",
"style": "rule"
} | {
"index": "35590",
"split": "train"
} |
math | [
{
"content": "Given a random variable $X \\sim B(10, 0.6)$, calculate the values of $E(X)$ and $D(X)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.4",
"style": "rule"
} | {
"index": "35591",
"split": "train"
} |
math | [
{
"content": "The average of 12, 21 and $x$ is 18. What is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "35592",
"split": "train"
} |
math | [
{
"content": "Let the sequence \\(b_1, b_2, b_3, \\dots\\) be defined such that \\(b_1 = 24\\), \\(b_{12} = 150\\), and for all \\(n \\geq 3\\), \\(b_n\\) is the arithmetic mean of the first \\(n - 1\\) terms. Find \\(b_2\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "276",
"style": "rule"
} | {
"index": "35593",
"split": "train"
} |
math | [
{
"content": "The numbers from 1 to 9 are placed in the cells of a \\(3 \\times 3\\) grid such that the sum of the numbers on one diagonal is 7 and on the other diagonal is 21. What is the sum of the numbers in the five shaded cells? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "35594",
"split": "train"
} |
math | [
{
"content": "Which are more: three-digit numbers where all digits have the same parity (all even or all odd), or three-digit numbers where adjacent digits have different parity? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "225",
"style": "rule"
} | {
"index": "35595",
"split": "train"
} |
math | [
{
"content": "Side $AB$ of triangle $ABC$ has length 8 inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is: Please output the final answer within \\b... | {
"ground_truth": "\\frac{40}{3}",
"style": "rule"
} | {
"index": "35596",
"split": "train"
} |
math | [
{
"content": "In the equilateral triangle \\(ABC\\), point \\(T\\) is its centroid, point \\(R\\) is the reflection of \\(T\\) across the line \\(AB\\), and point \\(N\\) is the reflection of \\(T\\) across the line \\(BC\\).\n\nDetermine the ratio of the areas of triangles \\(ABC\\) and \\(TRN\\). Please outpu... | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": "35597",
"split": "train"
} |
math | [
{
"content": "If 6 students want to sign up for 4 clubs, where students A and B do not join the same club, and every club must have at least one member with each student only joining one club, calculate the total number of different registration schemes. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "1320",
"style": "rule"
} | {
"index": "35598",
"split": "train"
} |
math | [
{
"content": "What is the sum of the squares of the lengths of the $\\textbf{medians}$ of a triangle whose side lengths are $10,$ $10,$ and $12$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "258",
"style": "rule"
} | {
"index": "35599",
"split": "train"
} |
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