data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In the numbers from 100 to 999, calculate how many numbers have digits in strictly increasing or strictly decreasing order. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "204",
"style": "rule"
} | {
"index": "31700",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal S$ be a set of $16$ points in the plane, no three collinear. Let $\\chi(S)$ denote the number of ways to draw $8$ lines with endpoints in $\\mathcal S$ , such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of $\\chi(\\mathcal S)$ ... | {
"ground_truth": "1430",
"style": "rule"
} | {
"index": "31701",
"split": "train"
} |
math | [
{
"content": "The line $y = 2x + c$ is tangent to the parabola $y^2 = 8x.$ Find $c.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "31702",
"split": "train"
} |
math | [
{
"content": "Given (1+i)x=1+yi, find |x+yi|. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "31703",
"split": "train"
} |
math | [
{
"content": "Consider a circle of radius 1 with center $O$ down in the diagram. Points $A$, $B$, $C$, and $D$ lie on the circle such that $\\angle AOB = 120^\\circ$, $\\angle BOC = 60^\\circ$, and $\\angle COD = 180^\\circ$. A point $X$ lies on the minor arc $\\overarc{AC}$. If $\\angle AXB = 90^\\circ$, find ... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "31704",
"split": "train"
} |
math | [
{
"content": "Given a tetrahedron \\(A B C D\\), where \\(B D = D C = C B = \\sqrt{2}\\), \\(A C = \\sqrt{3}\\), and \\(A B = A D = 1\\), find the cosine of the angle between line \\(B M\\) and line \\(A C\\), where \\(M\\) is the midpoint of \\(C D\\). Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "\\frac{\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "31705",
"split": "train"
} |
math | [
{
"content": "Given \\(\\frac{x+ \\sqrt{2}i}{i}=y+i\\), where \\(x\\), \\(y\\in\\mathbb{R}\\), and \\(i\\) is the imaginary unit, find \\(|x-yi|\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "31706",
"split": "train"
} |
math | [
{
"content": "Given $\\triangle ABC$ is an oblique triangle, with the lengths of the sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively. If $c\\sin A= \\sqrt {3}a\\cos C$.\n\n(Ⅰ) Find angle $C$; \n(Ⅱ) If $c= \\sqrt {21}$, and $\\sin C+\\sin (B-A)=5\\sin 2A$, find the area of $\\tr... | {
"ground_truth": "\\frac {5 \\sqrt {3}}{4}",
"style": "rule"
} | {
"index": "31707",
"split": "train"
} |
math | [
{
"content": "Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits of the integer $n$. For example, $P(23) = 6$ and $S(23) = 5$. Suppose $N$ is a two-digit number such that $N = P(N) +\nS(N)$. What is the units digit of $N$? Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "31708",
"split": "train"
} |
math | [
{
"content": "Given a trapezoid with labeled points as shown in the diagram, the point of intersection of the extended non-parallel sides is labeled as \\(E\\), and the point of intersection of the diagonals is labeled as \\(F\\). Similar right triangles \\(BFC\\) and \\(DFA\\) have corresponding sides \\(x, y\... | {
"ground_truth": "\\frac{4}{\\sqrt{3}}",
"style": "rule"
} | {
"index": "31709",
"split": "train"
} |
math | [
{
"content": "There are 200 computers in a computer center, some of which are connected by cables in pairs, with a total of 345 cables used. We call a \"cluster\" a set of computers such that any computer in this set can send a signal to all others through the cables. Initially, all computers formed one cluster... | {
"ground_truth": "153",
"style": "rule"
} | {
"index": "31710",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of all nondegenerate triangles formed from the vertices of a regular octagon with side length 1. Find the ratio of the largest area of any triangle in $S$ to the smallest area of any triangle in $S$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3+2 \\sqrt{2}",
"style": "rule"
} | {
"index": "31711",
"split": "train"
} |
math | [
{
"content": "Given the function f(x) = 3x^2 + 5x - 2, find the minimum value of the function f(x) in the interval [-2, -1]. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "31712",
"split": "train"
} |
math | [
{
"content": "A dark drawer contains $90$ red socks, $70$ green socks, $50$ blue socks, and $30$ purple socks. Someone randomly selects socks without seeing their color. What is the smallest number of socks that they must pick to guarantee at least $12$ pairs? A pair of socks consists of two socks of the same ... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "31713",
"split": "train"
} |
math | [
{
"content": "To address the threat of AIDS to humanity, now three research institutes, designated as Institute A, Institute B, and Institute C, are independently developing an AIDS vaccine. The probabilities of successfully developing a vaccine are respectively $\\frac{1}{2}$, $\\frac{1}{3}$, and $\\frac{1}{4}... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "31714",
"split": "train"
} |
math | [
{
"content": "Given M be the greatest five-digit number whose digits have a product of 36, determine the sum of the digits of M. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "31715",
"split": "train"
} |
math | [
{
"content": "Given that point \\( F \\) is the right focus of the ellipse \\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\) (\\(a > b > 0\\)), and the eccentricity of the ellipse is \\(\\frac{\\sqrt{3}}{2}\\), a line \\( l \\) passing through point \\( F \\) intersects the ellipse at points \\( A \\) and \\( B... | {
"ground_truth": "-\\sqrt{2}",
"style": "rule"
} | {
"index": "31716",
"split": "train"
} |
math | [
{
"content": "Students guess that Norb's age is $24, 28, 30, 32, 36, 38, 41, 44, 47$, and $49$. Norb says, \"At least half of you guessed too low, two of you are off by one, and my age is a prime number.\" How old is Norb? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "31717",
"split": "train"
} |
math | [
{
"content": "For how many integer values of $n$ between 1 and 120 inclusive does the decimal representation of $\\frac{n}{120}$ terminate? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "31718",
"split": "train"
} |
math | [
{
"content": "Two sides of a triangle are $8 \\mathrm{dm}$ and $5 \\mathrm{dm}$; the angle opposite to the first side is twice as large as the angle opposite to the second side. What is the length of the third side of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7.8",
"style": "rule"
} | {
"index": "31719",
"split": "train"
} |
math | [
{
"content": "How many integer values of $x$ satisfy $|x|<3\\pi$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "31720",
"split": "train"
} |
math | [
{
"content": "The minimum value of the sum of the distances from a point in space to the vertices of a regular tetrahedron with edge length 1 is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\sqrt{6}$",
"style": "rule"
} | {
"index": "31721",
"split": "train"
} |
math | [
{
"content": "A trapezoid \\(ABCD\\) (\\(AD \\parallel BC\\)) and a rectangle \\(A_{1}B_{1}C_{1}D_{1}\\) are inscribed in a circle \\(\\Omega\\) with a radius of 13 such that \\(AC \\parallel B_{1}D_{1}\\) and \\(BD \\parallel A_{1}C_{1}\\). Find the ratio of the areas of \\(ABCD\\) and \\(A_{1}B_{1}C_{1}D_{1}\... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31722",
"split": "train"
} |
math | [
{
"content": "A regular triangular prism $ABC A_{1} B_{1} C_{1}$ with the base $ABC$ and lateral edges $A A_{1}, B B_{1}, C C_{1}$ is inscribed in a sphere of radius 3. Segment $CD$ is a diameter of this sphere. Find the volume of the prism if $A D = 2 \\sqrt{6}$. Please output the final answer within \\boxed{}... | {
"ground_truth": "6\\sqrt{15}",
"style": "rule"
} | {
"index": "31723",
"split": "train"
} |
math | [
{
"content": "What number must we add to $3-5i$ to get $2+7i$? Here, $i^2 = -1.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1+12i",
"style": "rule"
} | {
"index": "31724",
"split": "train"
} |
math | [
{
"content": "A relatively prime date is defined as a date where the day and the month number are coprime. Determine how many relatively prime dates are in the month with 31 days and the highest number of non-relatively prime dates? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31725",
"split": "train"
} |
math | [
{
"content": "Given that the sum of the first $n$ terms of the arithmetic sequence ${a_n}$ is $S_n$, if $a_2=0$, $S_3+S_4=6$, then the value of $a_5+a_6$ is $\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "31726",
"split": "train"
} |
math | [
{
"content": "The graphs $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have y-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer x-intercepts. Find $h$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "31727",
"split": "train"
} |
math | [
{
"content": "A club has 99 members. Find the smallest positive integer $n$ such that if the number of acquaintances of each person is greater than $n$, there must exist 4 people who all know each other mutually (here it is assumed that if $A$ knows $B$, then $B$ also knows $A$). Please output the final answer ... | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "31728",
"split": "train"
} |
math | [
{
"content": "Inside a cube \\( C \\) with side length 1, an inscribed sphere \\( O_{1} \\) is placed. Another sphere \\( O_{2} \\) is also placed inside \\( C \\) such that it is tangent to the inscribed sphere \\( O_{1} \\) and also tangent to three faces of the cube. What is the surface area of sphere \\( O_... | {
"ground_truth": "(7 - 4\\sqrt{3})\\pi",
"style": "rule"
} | {
"index": "31729",
"split": "train"
} |
math | [
{
"content": "Find at the point \\( M_{0}(1,1,1) \\) the direction of the greatest change of the scalar field \\( u = xy + yz + xz \\) and the magnitude of this greatest change at this point. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "31730",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_1=1$ and $a_{n+1}=2S_n+1$, where $S_n$ is the sum of the first $n$ terms of $\\{a_n\\}$, and $n \\in \\mathbb{N}^*$.\n\n$(1)$ Find $a_n$.\n\n$(2)$ If the sequence $\\{b_n\\}$ satisfies $b_n=\\frac{1}{(1+\\log_3{a_n})(3+\\log_3{a_n})}$, and the sum of ... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "31731",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = a\\cos x - \\left(x - \\frac{\\pi}{2}\\right)\\sin x$, where $x \\in \\left[0, \\frac{\\pi}{2}\\right]$. \n$(1)$ When $a = -1$, find the range of the derivative ${f'}(x)$ of the function $f(x)$. \n$(2)$ If $f(x) \\leq 0$ always holds, find the maximum value of the real number $a$. Pl... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "31732",
"split": "train"
} |
math | [
{
"content": "If 554 is the base $b$ representation of the square of the number whose base $b$ representation is 24, then $b$, when written in base 10, equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "31733",
"split": "train"
} |
math | [
{
"content": "Find the number of pairs of integers $(x, y)$ such that $x^{2}+2y^{2}<25$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "31734",
"split": "train"
} |
math | [
{
"content": "How many integers from 1 to 1997 have a sum of digits that is divisible by 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "399",
"style": "rule"
} | {
"index": "31735",
"split": "train"
} |
math | [
{
"content": "A flag in the shape of a square has a symmetrical cross of uniform width, featuring a center decorated with a green square, and set against a yellow background. The entire cross (comprising both the arms and the green center) occupies 49% of the total area of the flag. What percentage of the flag'... | {
"ground_truth": "25\\%",
"style": "rule"
} | {
"index": "31736",
"split": "train"
} |
math | [
{
"content": "A bacterium develops on a $100 \\times 100$ grid. It can contaminate a new cell if and only if two adjacent cells are already contaminated. What is the minimal number of initially contaminated cells required for the bacterium to be able to spread everywhere on the grid? Please output the final ans... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "31737",
"split": "train"
} |
math | [
{
"content": "If the function $f\\left(x\\right)=\\frac{1}{2}\\left(m-2\\right){x}^{2}+\\left(n-8\\right)x+1\\left(m\\geqslant 0,n\\geqslant 0\\right)$ is monotonically decreasing in the interval $\\left[\\frac{1}{2},2\\right]$, find the maximum value of $mn$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "31738",
"split": "train"
} |
math | [
{
"content": "The specific heat capacity of a body with mass $m=2$ kg depends on the temperature as follows: $c=c_{0}(1+\\alpha t)$, where $c_{0}=150$ J/kg$\\cdot^\\circ$C is the specific heat capacity at $0^\\circ$C, $\\alpha=0.05\\ ^\\circ$C$^{-1}$ is the temperature coefficient, and $t$ is the temperature in... | {
"ground_truth": "96000",
"style": "rule"
} | {
"index": "31739",
"split": "train"
} |
math | [
{
"content": "Given the function \\( f(n) \\) defined on the set of natural numbers \\(\\mathbf{N}\\), and satisfies:\n\n\\[\n\\begin{array}{l}\nf(1) = f(2) = 1, \\\\\nf(3n) = 3 f(n) - 2, \\\\\nf(3n+1) = 3 f(n) + 1, \\\\\nf(3n+2) = 3 f(n) + 4 \\quad (n \\in \\mathbf{N}).\n\\end{array}\n\\]\n\nDetermine the larg... | {
"ground_truth": "1093",
"style": "rule"
} | {
"index": "31740",
"split": "train"
} |
math | [
{
"content": "Divide 2 teachers and 6 students into 2 groups, each group consisting of 1 teacher and 3 students. Calculate the total number of different arrangements to assign them to locations A and B to participate in social practice activities. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "31741",
"split": "train"
} |
math | [
{
"content": "After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average of these $36$ numbers. The ratio of the second average to the true average was Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1:1",
"style": "rule"
} | {
"index": "31742",
"split": "train"
} |
math | [
{
"content": "Find all real numbers $k$ for which there exists a nonzero, 2-dimensional vector $\\mathbf{v}$ such that\n\\[\\begin{pmatrix} 1 & 8 \\\\ 2 & 1 \\end{pmatrix} \\mathbf{v} = k \\mathbf{v}.\\]Enter all the solutions, separated by commas. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "31743",
"split": "train"
} |
math | [
{
"content": "Floyd looked at a standard $12$ hour analogue clock at $2\\!:\\!36$ . When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3\\!:\\!00$ was that? Please output the final answer withi... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "31744",
"split": "train"
} |
math | [
{
"content": "The inclination angle of the line $x- \\sqrt {3}y+3=0$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {\\pi}{6}",
"style": "rule"
} | {
"index": "31745",
"split": "train"
} |
math | [
{
"content": "Radford and Peter ran a race, during which they both ran at a constant speed. Radford began the race 30 m ahead of Peter. After 3 minutes, Peter was 18 m ahead of Radford. Peter won the race exactly 7 minutes after it began. How far from the finish line was Radford when Peter won? Please output th... | {
"ground_truth": "82 \\mathrm{~m}",
"style": "rule"
} | {
"index": "31746",
"split": "train"
} |
math | [
{
"content": "The hypotenuse of a right triangle is $10$ inches and the radius of the inscribed circle is $1$ inch. The perimeter of the triangle in inches is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "31747",
"split": "train"
} |
math | [
{
"content": "Points $P$ and $Q$ are midpoints of two sides of the square. What fraction of the interior of the square is shaded? Express your answer as a common fraction.\n\n[asy]\nfilldraw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--cycle,gray,linewidth(1));\nfilldraw((0,1)--(1,2)--(2,2)--(0,1)--cycle,white,linewidth(... | {
"ground_truth": "\\frac{7}{8}",
"style": "rule"
} | {
"index": "31748",
"split": "train"
} |
math | [
{
"content": "Given a square $R_1$ with an area of 25, where each side is trisected to form a smaller square $R_2$, and this process is repeated to form $R_3$ using the same trisection points strategy, calculate the area of $R_3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{400}{81}",
"style": "rule"
} | {
"index": "31749",
"split": "train"
} |
math | [
{
"content": "What is $\\frac{~\\frac{2}{5}~}{\\frac{3}{7}}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{14}{15}",
"style": "rule"
} | {
"index": "31750",
"split": "train"
} |
math | [
{
"content": "The inclination angle of the line $x-y+1=0$ is what value? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "31751",
"split": "train"
} |
math | [
{
"content": "\\(ABCD\\) is a parallelogram with \\(AB = 7\\), \\(BC = 2\\), and \\(\\angle DAB = 120^\\circ\\). Parallelogram \\(ECFA\\) is contained within \\(ABCD\\) and is similar to it. Find the ratio of the area of \\(ECFA\\) to the area of \\(ABCD\\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "39/67",
"style": "rule"
} | {
"index": "31752",
"split": "train"
} |
math | [
{
"content": "Let $f$ be the function defined by $f(x) = -2 \\sin(\\pi x)$. How many values of $x$ such that $-2 \\le x \\le 2$ satisfy the equation $f(f(f(x))) = f(x)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "61",
"style": "rule"
} | {
"index": "31753",
"split": "train"
} |
math | [
{
"content": "A certain store sells a product, and because the purchase price decreased by 6.4% compared to the original purchase price, the profit margin increased by 8 percentage points. What was the original profit margin for selling this product? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "17\\%",
"style": "rule"
} | {
"index": "31754",
"split": "train"
} |
math | [
{
"content": "A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\\lfloor D\\rfloor$? (For rea... | {
"ground_truth": "947",
"style": "rule"
} | {
"index": "31755",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ $c$ be distinct complex numbers such that\n\\[\\frac{a}{1 - b} = \\frac{b}{1 - c} = \\frac{c}{1 - a} = k.\\]Find the sum of all possible values of $k.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "31756",
"split": "train"
} |
math | [
{
"content": "If $\\sqrt{2+\\sqrt x}=3$, what is $x?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "31757",
"split": "train"
} |
math | [
{
"content": "Given 500 points inside a convex 1000-sided polygon, along with the polygon's vertices (a total of 1500 points), none of which are collinear, the polygon is divided into triangles with these 1500 points as the vertices of the triangles. There are no other vertices apart from these. How many triang... | {
"ground_truth": "1998",
"style": "rule"
} | {
"index": "31758",
"split": "train"
} |
math | [
{
"content": "A two-digit number is divided by the sum of its digits. The result is a number between 2.6 and 2.7. Find all of the possible values of the original two-digit number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "31759",
"split": "train"
} |
math | [
{
"content": "Convert $115_{10}$ to base 11. Represent $10$ as $A$, if necessary. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\text{A5}_{11}",
"style": "rule"
} | {
"index": "31760",
"split": "train"
} |
math | [
{
"content": "The value of $2x+10$ is half of the value of $5x+30$. What is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-10",
"style": "rule"
} | {
"index": "31761",
"split": "train"
} |
math | [
{
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $(x^{2}- \\dfrac {y^{2}}{24}=1)$, and $P$ is a point on the hyperbola, find the area of $\\triangle F\\_1PF\\_2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "31762",
"split": "train"
} |
math | [
{
"content": "Remove five out of twelve digits so that the remaining numbers sum up to 1111.\n\n$$\n\\begin{array}{r}\n111 \\\\\n333 \\\\\n+\\quad 777 \\\\\n999 \\\\\n\\hline 1111\n\\end{array}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1111",
"style": "rule"
} | {
"index": "31763",
"split": "train"
} |
math | [
{
"content": "In a positive term geometric sequence ${a_n}$, ${a_5 a_6 =81}$, calculate the value of ${\\log_{3}{a_1} + \\log_{3}{a_5} +...+\\log_{3}{a_{10}}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "31764",
"split": "train"
} |
math | [
{
"content": "If 1983 were expressed as a sum of distinct powers of 2, with the requirement that at least five distinct powers of 2 are used, what would be the least possible sum of the exponents of these powers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "31765",
"split": "train"
} |
math | [
{
"content": "The product of two of the four roots of the quartic equation \\( x^4 - 18x^3 + kx^2 + 200x - 1984 = 0 \\) is -32. Determine the value of \\( k \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "86",
"style": "rule"
} | {
"index": "31766",
"split": "train"
} |
math | [
{
"content": "The inhabitants of the island of Jumble use the standard Kobish alphabet ($20$ letters, A through T). Each word in their language is $4$ letters or less, and for some reason, they insist that all words contain the letter A at least once. How many words are possible? Please output the final answer... | {
"ground_truth": "30860",
"style": "rule"
} | {
"index": "31767",
"split": "train"
} |
math | [
{
"content": "Let the odd function $f(x)$ defined on $\\mathbb{R}$ satisfy $f(x+3) = -f(1-x)$. If $f(3) = 2$, then $f(2013) = \\ $. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "31768",
"split": "train"
} |
math | [
{
"content": "Given a triangular pyramid \\( S-ABC \\) whose base \\( ABC \\) is an isosceles right triangle with \\( AB \\) as the hypotenuse, and satisfying \\( SA = SB = SC = 2 \\) and \\( AB = 2 \\), assume that the four points \\( S, A, B, C \\) are all on the surface of a sphere centered at point \\( O \\... | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "31769",
"split": "train"
} |
math | [
{
"content": "The digits $1,2,3,4,5,$ and $6$ can be arranged to form many different $6$-digit positive integers with six distinct digits. In how many such integers is the digit $1$ to the left of both the digits $2$ and $3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "31770",
"split": "train"
} |
math | [
{
"content": "Let $(2x+1)^6 = a_0x^6 + a_1x^5 + a_2x^4 + a_3x^3 + a_4x^2 + a_5x + a_6$, which is an identity in $x$ (i.e., it holds for any $x$). Try to find the values of the following three expressions: \n(1) $a_0 + a_1 + a_2 + a_3 + a_4 + a_5 + a_6$; \n(2) $a_1 + a_3 + a_5$; \n(3) $a_2 + a_4$. Please outp... | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "31771",
"split": "train"
} |
math | [
{
"content": "Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2028",
"style": "rule"
} | {
"index": "31772",
"split": "train"
} |
math | [
{
"content": "There is a box containing 3 red balls and 3 white balls, which are identical in size and shape. A fair die is rolled, and the number rolled determines the number of balls drawn from the box. What is the probability that the number of red balls drawn is greater than the number of white balls drawn?... | {
"ground_truth": "19/60",
"style": "rule"
} | {
"index": "31773",
"split": "train"
} |
math | [
{
"content": "Calculate the arc lengths of curves given by the equations in polar coordinates.\n\n$$\n\\rho = 3 \\varphi, \\quad 0 \\leq \\varphi \\leq \\frac{4}{3}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10}{3} + \\frac{3}{2} \\ln 3",
"style": "rule"
} | {
"index": "31774",
"split": "train"
} |
math | [
{
"content": "The diagonal lengths of a rhombus are 24 units and 10 units. What is the area of the rhombus, in square units? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "31775",
"split": "train"
} |
math | [
{
"content": "Ana and Luíza train every day for the Big Race that will take place at the end of the year at school, each running at the same speed. The training starts at point $A$ and ends at point $B$, which are $3000 \\mathrm{~m}$ apart. They start at the same time, but when Luíza finishes the race, Ana stil... | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "31776",
"split": "train"
} |
math | [
{
"content": "The scientific notation for 0.000048 is $4.8\\times 10^{-5}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.8 \\times 10^{-5}",
"style": "rule"
} | {
"index": "31777",
"split": "train"
} |
math | [
{
"content": "An amusement park has a series of miniature buildings and landscape models from various places in the United States, scaled at $1: 20$. If the height of the United States Capitol Building is 289 feet, what is the nearest integer height of its model in feet? Please output the final answer within \\... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "31778",
"split": "train"
} |
math | [
{
"content": "The diagram shows a solid with six triangular faces and five vertices. Andrew wants to write an integer at each of the vertices so that the sum of the numbers at the three vertices of each face is the same. He has already written the numbers 1 and 5 as shown. What is the sum of the other three num... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31779",
"split": "train"
} |
math | [
{
"content": "Find $120_4\\times13_4\\div2_4$. Express your answer in base 4. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1110_4",
"style": "rule"
} | {
"index": "31780",
"split": "train"
} |
math | [
{
"content": "There are three pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ that satisfy both $x^3 - 3xy^2 = 2010$ and $y^3 - 3x^2y = 2000$. Compute $\\left(1-\\frac{x_1}{y_1}\\right)\\left(1-\\frac{x_2}{y_2}\\right)\\left(1-\\frac{x_3}{y_3}\\right)$. Please output the final answer within \\bo... | {
"ground_truth": "\\frac{1}{100}",
"style": "rule"
} | {
"index": "31781",
"split": "train"
} |
math | [
{
"content": "On a table, there are 10 cards numbered $1, 1, 2, 2, 3, 3, 4, 4, 5, 5$. These 10 cards are shuffled and arranged in a row from left to right. Then, the number of cards between the two 1s, the two 2s, the two 3s, the two 4s, and the two 5s are counted. What is the maximum sum of these 5 numbers? Pl... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "31782",
"split": "train"
} |
math | [
{
"content": "Let $(2-x)^{6}=a_{0}+a_{1}x+a_{2}x^{2}+\\ldots+a_{6}x^{6}$, then the value of $|a_{1}|+|a_{2}|+\\ldots+|a_{6}|$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "665",
"style": "rule"
} | {
"index": "31783",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer $n$ such that $7350$ is a factor of $n!$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31784",
"split": "train"
} |
math | [
{
"content": "Each of the $39$ students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and $26$ students have a cat. How many students have both a dog and a cat? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "31785",
"split": "train"
} |
math | [
{
"content": "Find the volume of the region in space defined by\n\\[|x + 2y + z| + |x + 2y - z| \\le 12\\]\nand \\(x, y, z \\ge 0.\\) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "31786",
"split": "train"
} |
math | [
{
"content": "Given that the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is $60^{\\circ}$, $|\\overrightarrow{a}|=1$, and $|\\overrightarrow{b}|=2$,\n(1) Find $(2\\overrightarrow{a}-\\overrightarrow{b})\\cdot\\overrightarrow{a}$;\n(2) Find $|2\\overrightarrow{a}+\\overrightarrow{b}|$. ... | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "31787",
"split": "train"
} |
math | [
{
"content": "Given that the random variable $\\xi$ follows a normal distribution $N(2, \\sigma^2)$, and $P(\\xi < 3) = 0.6$, find $P(1 < \\xi < 2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.1",
"style": "rule"
} | {
"index": "31788",
"split": "train"
} |
math | [
{
"content": "The sequence starts at 2,187,000 and each subsequent number is created by dividing the previous number by 3. What is the last integer in this sequence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": "31789",
"split": "train"
} |
math | [
{
"content": "A bicycle factory plans to produce a batch of bicycles of the same model, planning to produce $220$ bicycles per day. However, due to various reasons, the actual daily production will differ from the planned quantity. The table below shows the production situation of the workers in a certain week:... | {
"ground_truth": "155080",
"style": "rule"
} | {
"index": "31790",
"split": "train"
} |
math | [
{
"content": "Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respective midpoints of $A B, A C, A D, B C, B D, C D$. If $Q R=2001, S U=2002, T V=$ 2003, find the distance between the midpoints of $Q U$ and $R V$. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "2001",
"style": "rule"
} | {
"index": "31791",
"split": "train"
} |
math | [
{
"content": "$A B C D$ is a regular tetrahedron of volume 1. Maria glues regular tetrahedra $A^{\\prime} B C D, A B^{\\prime} C D$, $A B C^{\\prime} D$, and $A B C D^{\\prime}$ to the faces of $A B C D$. What is the volume of the tetrahedron $A^{\\prime} B^{\\prime} C^{\\prime} D^{\\prime}$? Please output the ... | {
"ground_truth": "\\frac{125}{27}",
"style": "rule"
} | {
"index": "31792",
"split": "train"
} |
math | [
{
"content": "Determine the value of the infinite product $(3^{1/4})(9^{1/16})(27^{1/64})(81^{1/256}) \\dotsm$ plus 2, the result in the form of \"$\\sqrt[a]{b}$ plus $c$\". Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt[9]{81} + 2",
"style": "rule"
} | {
"index": "31793",
"split": "train"
} |
math | [
{
"content": "Two vertices of an obtuse triangle are $(6,4)$ and $(0,0)$. The third vertex is located on the negative branch of the $x$-axis. What are the coordinates of the third vertex if the area of the triangle is 30 square units? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-15, 0)",
"style": "rule"
} | {
"index": "31794",
"split": "train"
} |
math | [
{
"content": "How many functions $f$ from \\{-1005, \\ldots, 1005\\} to \\{-2010, \\ldots, 2010\\} are there such that the following two conditions are satisfied? - If $a<b$ then $f(a)<f(b)$. - There is no $n$ in \\{-1005, \\ldots, 1005\\} such that $|f(n)|=|n|$ Please output the final answer within \\boxed{}."... | {
"ground_truth": "1173346782666677300072441773814388000553179587006710786401225043842699552460942166630860 5302966355504513409792805200762540756742811158611534813828022157596601875355477425764387 2333935841666957750009216404095352456877594554817419353494267665830087436353494075828446 00705064877936286986176650915007... | {
"index": "31795",
"split": "train"
} |
math | [
{
"content": "What is the probability that the same number will be facing up on each of three standard six-sided dice that are tossed simultaneously? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{36}",
"style": "rule"
} | {
"index": "31796",
"split": "train"
} |
math | [
{
"content": "Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain? Please output... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "31797",
"split": "train"
} |
math | [
{
"content": "A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "31798",
"split": "train"
} |
math | [
{
"content": "Given that $a$ and $b$ are real numbers, and $\\frac{a}{1-i} + \\frac{b}{2-i} = \\frac{1}{3-i}$, find the sum of the first 100 terms of the arithmetic sequence ${an + b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-910",
"style": "rule"
} | {
"index": "31799",
"split": "train"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.