data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Points $F$, $G$, $H$, $I$, and $J$ are positioned in 3-dimensional space with $FG = GH = HI = IJ = JF = 3$ and $\\angle FGH = \\angle HIJ = \\angle JIF = 90^\\circ$. The plane of triangle $FGH$ is parallel to $\\overline{IJ}$. Determine the area of triangle $GIJ$. Please output the final answer wi... | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "31500",
"split": "train"
} |
math | [
{
"content": "What is the product of all real numbers that are tripled when added to their reciprocals? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "31501",
"split": "train"
} |
math | [
{
"content": "Consider all prisms whose base is a convex 2015-gon.\n\nWhat is the maximum number of edges of such a prism that can be intersected by a plane not passing through its vertices? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2017",
"style": "rule"
} | {
"index": "31502",
"split": "train"
} |
math | [
{
"content": "The increasing sequence $3, 15, 24, 48, \\ldots\\,$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "31503",
"split": "train"
} |
math | [
{
"content": "Circles $A,B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius 1 and passes through the center of $D$. What is the radius of circle $B$?\n\n[asy]unitsize(1cm);\npair A,B,C,D;\nA=(-1,0);\nB=(0.66,0.88);\nC=(... | {
"ground_truth": "\\frac{8}{9}",
"style": "rule"
} | {
"index": "31504",
"split": "train"
} |
math | [
{
"content": "Given that $\\sum_{k=1}^{35}\\sin 5k=\\tan \\frac mn,$ where angles are measured in degrees, and $m_{}$ and $n_{}$ are relatively prime positive integers that satisfy $\\frac mn<90,$ find $m+n.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "177",
"style": "rule"
} | {
"index": "31505",
"split": "train"
} |
math | [
{
"content": "Consider a 5-minute interval. In this period, an average of 5 bites occur on the first fishing rod, and 1 bite on the second fishing rod. Therefore, the total average number of bites on both rods during these 5 minutes is 6. Determine the average waiting time for the first bite. Please output the ... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "31506",
"split": "train"
} |
math | [
{
"content": "If the lengths of the sides of a triangle are positive integers not greater than 5, how many such distinct triangles exist? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "31507",
"split": "train"
} |
math | [
{
"content": "Given that the slant height of a certain cone is $4$ and the height is $2\\sqrt{3}$, calculate the total surface area of the cone. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12\\pi",
"style": "rule"
} | {
"index": "31508",
"split": "train"
} |
math | [
{
"content": "A right triangle XYZ has legs XY = YZ = 8 cm. In each step of an iterative process, the triangle is divided into four smaller right triangles by joining the midpoints of the sides. However, for this problem, the area of the shaded triangle in each iteration is reduced by a factor of 3 rather than ... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "31509",
"split": "train"
} |
math | [
{
"content": "The volume of the parallelepiped determined by vectors $\\mathbf{a}$, $\\mathbf{b}$, and $\\mathbf{c}$ is 6. Find the volume of the parallelepiped determined by $\\mathbf{a} + 2\\mathbf{b},$ $\\mathbf{b} + \\mathbf{c},$ and $2\\mathbf{c} - 5\\mathbf{a}.$ Please output the final answer within \\box... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "31510",
"split": "train"
} |
math | [
{
"content": "Let $d$ be a positive number such that when $144$ is divided by $d$, the remainder is $9$. Compute the sum of all possible two-digit values of $d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "87",
"style": "rule"
} | {
"index": "31511",
"split": "train"
} |
math | [
{
"content": "Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{23}{108}",
"style": "rule"
} | {
"index": "31512",
"split": "train"
} |
math | [
{
"content": "A jar is filled with red, orange and yellow jelly beans. The probability of randomly selecting a red jelly bean from this jar is $0.2$, and the probability of randomly selecting an orange jelly bean from this jar is $0.5$. What is the probability of randomly selecting a yellow jelly bean from th... | {
"ground_truth": "0.3",
"style": "rule"
} | {
"index": "31513",
"split": "train"
} |
math | [
{
"content": "In a store, there are 21 white shirts and 21 purple shirts hanging in a row. Find the minimum number \\( k \\) such that, regardless of the initial order of the shirts, it is possible to take down \\( k \\) white shirts and \\( k \\) purple shirts so that the remaining white shirts are all hanging... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31514",
"split": "train"
} |
math | [
{
"content": "Given \\( f_{1}(x)=-\\frac{2x+7}{x+3}, \\) and \\( f_{n+1}(x)=f_{1}(f_{n}(x)), \\) for \\( x \\neq -2, x \\neq -3 \\), find the value of \\( f_{2022}(2021) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2021",
"style": "rule"
} | {
"index": "31515",
"split": "train"
} |
math | [
{
"content": "Regular octagon $C H I L D R E N$ has area 1. Find the area of pentagon $C H I L D$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31516",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered integer pairs \\((a, b)\\) such that the equation \\(x^{2} + a x + b = 167y\\) has integer solutions \\((x, y)\\), where \\(1 \\leq a, b \\leq 2004\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2020032",
"style": "rule"
} | {
"index": "31517",
"split": "train"
} |
math | [
{
"content": "In triangle \\( ABC \\) with sides \\( AB = 8 \\), \\( AC = 4 \\), \\( BC = 6 \\), the angle bisector \\( AK \\) is drawn, and on the side \\( AC \\) a point \\( M \\) is marked such that \\( AM : CM = 3 : 1 \\). Point \\( N \\) is the intersection point of \\( AK \\) and \\( BM \\). Find \\( AN \... | {
"ground_truth": "\\frac{18\\sqrt{6}}{11}",
"style": "rule"
} | {
"index": "31518",
"split": "train"
} |
math | [
{
"content": "Given the line $l: x-y+4=0$ and the circle $C: \\begin{cases}x=1+2\\cos \\theta \\\\ y=1+2\\sin \\theta\\end{cases} (\\theta$ is a parameter), find the distance from each point on $C$ to $l$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\sqrt{2}-2",
"style": "rule"
} | {
"index": "31519",
"split": "train"
} |
math | [
{
"content": "In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, point $E$ is on $A_{1} D_{1}$, point $F$ is on $C D$, and $A_{1} E = 2 E D_{1}$, $D F = 2 F C$. Find the volume of the triangular prism $B-F E C_{1}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{27}",
"style": "rule"
} | {
"index": "31520",
"split": "train"
} |
math | [
{
"content": "Given $\\cos(\\alpha + \\frac{\\pi}{4}) = \\frac{\\sqrt{2}}{4}$, find the value of $\\sin(2\\alpha)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "31521",
"split": "train"
} |
math | [
{
"content": "A rectangle has dimensions $4$ and $2\\sqrt{3}$. Two equilateral triangles are contained within this rectangle, each with one side coinciding with the longer side of the rectangle. The triangles intersect, forming another polygon. What is the area of this polygon?\nA) $2\\sqrt{3}$\nB) $4\\sqrt{3}$... | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "31522",
"split": "train"
} |
math | [
{
"content": "Find the positive value of $x$ which satisfies\n\\[\\log_5 (x - 2) + \\log_{\\sqrt{5}} (x^3 - 2) + \\log_{\\frac{1}{5}} (x - 2) = 4.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "31523",
"split": "train"
} |
math | [
{
"content": "Given an isosceles triangle \\(XYZ\\) with \\(XY = YZ\\) and an angle at the vertex equal to \\(96^{\\circ}\\). Point \\(O\\) is located inside triangle \\(XYZ\\) such that \\(\\angle OZX = 30^{\\circ}\\) and \\(\\angle OXZ = 18^{\\circ}\\). Find the measure of angle \\(\\angle YOX\\). Please outp... | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "31524",
"split": "train"
} |
math | [
{
"content": "The sides of a triangle have lengths \\( 13, 17, \\) and \\( k, \\) where \\( k \\) is a positive integer. For how many values of \\( k \\) is the triangle obtuse? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "31525",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence ${a_{n}}$, let $S_{n}$ denote the sum of its first $n$ terms. The first term $a_{1}$ is given as $-20$. The common difference is a real number in the interval $(3,5)$. Determine the probability that the minimum value of $S_{n}$ is only $S_{6}$. Please output the final ... | {
"ground_truth": "\\dfrac{1}{3}",
"style": "rule"
} | {
"index": "31526",
"split": "train"
} |
math | [
{
"content": "The fraction \\(\\frac{1}{221}\\) is represented as a repeating decimal. Find the length of the repeating sequence. (For example, the length of the repeating sequence for the fraction \\(\\frac{25687}{99900} = 0.25712712712 \\ldots = 0.25(712)\\) is 3.) Please output the final answer within \\boxe... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "31527",
"split": "train"
} |
math | [
{
"content": "The value of x that satisfies the equation \\( x^{x^x} = 2 \\) is calculated. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "31528",
"split": "train"
} |
math | [
{
"content": "In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\\angle A = 37^\\circ$, $\\angle D = 53^\\circ$, and $M$ and $N$ be the midpoints of $\\overline{BC}$ and $\\overline{AD}$, respectively. Find the length $MN$. Please output the final answer wi... | {
"ground_truth": "504",
"style": "rule"
} | {
"index": "31529",
"split": "train"
} |
math | [
{
"content": "Let \\( S = \\{1, 2, \\ldots, 98\\} \\). Find the smallest natural number \\( n \\) such that from any subset of \\( S \\) containing \\( n \\) elements, we can always select 10 numbers. No matter how we divide these 10 numbers into two groups, there will always be one group containing a number th... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "31530",
"split": "train"
} |
math | [
{
"content": "The total price of 4 floppy disks and 5 CDs is not less than 20 yuan, and the total price of 6 floppy disks and 3 CDs is not more than 24 yuan. What is the minimum amount of money needed to buy 3 floppy disks and 9 CDs? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "31531",
"split": "train"
} |
math | [
{
"content": "If $3+a=4-b$ and $4+b=7+a$, what is $3-a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "31532",
"split": "train"
} |
math | [
{
"content": "Calculate the largest prime factor of $18^4 + 12^5 - 6^6$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31533",
"split": "train"
} |
math | [
{
"content": "Given that $F$ is the right focus of the hyperbola $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, and the line $l$ passing through the origin intersects the hyperbola at points $M$ and $N$, with $\\overrightarrow{MF} \\cdot \\overrightarrow{NF} = 0$. If the area of $\\triangle M... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "31534",
"split": "train"
} |
math | [
{
"content": "If $x@y=xy-2x$, what is the value of $(7@4)-(4@7)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "31535",
"split": "train"
} |
math | [
{
"content": "A car travels 40 kph for 20 kilometers, 50 kph for 25 kilometers, 60 kph for 45 minutes and 48 kph for 15 minutes. What is the average speed of the car, in kph? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "31536",
"split": "train"
} |
math | [
{
"content": "Calculate the expression $200(200-5)-(200\\cdot200-5)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-995",
"style": "rule"
} | {
"index": "31537",
"split": "train"
} |
math | [
{
"content": "Each successive term in the sequence $2048, 512, 128, x,\ny, 2, \\frac 12, \\frac 18,...$ is obtained by multiplying the previous term by a constant. What is the value of $x + y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "31538",
"split": "train"
} |
math | [
{
"content": "Two balls, a red one and a green one, are randomly and independently tossed into bins numbered with positive integers. For each ball, the probability that it is tossed into bin $k$ is given by $p_k = \\frac{1}{k(k+1)}$ for $k = 1, 2, 3, ...$. Calculate the probability that the red ball is tossed i... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "31539",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC, AB = 10, BC = 8, CA = 7$ and side $BC$ is extended to a point $P$ such that $\\triangle PAB$ is similar to $\\triangle PCA$. Calculate the length of $PC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{56}{3}",
"style": "rule"
} | {
"index": "31540",
"split": "train"
} |
math | [
{
"content": "Let $z$ be a complex number with $|z|=2014$. Let $P$ be the polygon in the complex plane whose vertices are $z$ and every $w$ such that $\\frac{1}{z+w}=\\frac{1}{z}+\\frac{1}{w}$. Then the area enclosed by $P$ can be written in the form $n\\sqrt{3}$, where $n$ is an integer. Find the remainder whe... | {
"ground_truth": "147",
"style": "rule"
} | {
"index": "31541",
"split": "train"
} |
math | [
{
"content": "The maximum and minimum values of the function $f(x) = -x^2 + 2x + 3$ in the interval $[-2, 3]$ are ___ and ___, respectively. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "31542",
"split": "train"
} |
math | [
{
"content": "After expanding and combining like terms in the expression \\((x+y+z)^{2028} + (x-y-z)^{2028}\\), how many monomials of the form \\(x^{a} y^{b} z^{c}\\) have a non-zero coefficient? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1030225",
"style": "rule"
} | {
"index": "31543",
"split": "train"
} |
math | [
{
"content": "Find the largest value of $n$ such that $5x^2+nx+48$ can be factored as the product of two linear factors with integer coefficients. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "241",
"style": "rule"
} | {
"index": "31544",
"split": "train"
} |
math | [
{
"content": "A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total co... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31545",
"split": "train"
} |
math | [
{
"content": "Find the fraction with the smallest possible denominator that lies between \\(\\frac{7}{25}\\) and \\(\\frac{8}{25}\\). Provide all possible solutions. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{7}",
"style": "rule"
} | {
"index": "31546",
"split": "train"
} |
math | [
{
"content": "When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of five consecutive positive integers all of which are nonprime? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "31547",
"split": "train"
} |
math | [
{
"content": "Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\\leq i < j < k\\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle. Please output the final answ... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "31548",
"split": "train"
} |
math | [
{
"content": "Let $p$ and $q$ be constants. Suppose that the equation \\[\\frac{(x+p)(x+q)(x+20)}{(x+4)^2} = 0\\] has exactly $3$ distinct roots, while the equation \\[\\frac{(x+3p)(x+4)(x+10)}{(x+q)(x+20)} = 0\\] has exactly $1$ distinct root. Compute $100p + q.$ Please output the final answer within \\boxed{}... | {
"ground_truth": "\\frac{430}{3}",
"style": "rule"
} | {
"index": "31549",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$.\n$\\textbf{(A)}\\ 4\\qquad \\textbf{(B)}\\ \\sqrt{17} \\qquad \\textbf{(C)}\\ 4.25\\qquad \\textbf{(D)}\\ 2\\sqrt{5} \\qquad \\textbf{(E)}\\ 4.5$ Please output the ... | {
"ground_truth": "\\sqrt{17}",
"style": "rule"
} | {
"index": "31550",
"split": "train"
} |
math | [
{
"content": "Compute $\\dbinom{9}{2}\\times \\dbinom{7}{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "756",
"style": "rule"
} | {
"index": "31551",
"split": "train"
} |
math | [
{
"content": "For all \\( m \\) and \\( n \\) satisfying \\( 1 \\leq n \\leq m \\leq 5 \\), the polar equation\n$$\n\\rho = \\frac{1}{1 - C_{m}^{n} \\cos \\theta}\n$$\nrepresents how many different hyperbolas? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31552",
"split": "train"
} |
math | [
{
"content": "If $547\\,932\\,BC4$ is divisible by $12$, where $B$ and $C$ represent digits, what is the sum of all unique values of $B + C$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "31553",
"split": "train"
} |
math | [
{
"content": "It is known that Xiao Li has a probability of hitting the bullseye of $40\\%$ each time he shoots. Now, the method of random simulation is used to estimate the probability that Xiao Li hits the bullseye exactly twice in three shots. First, a calculator is used to generate random integers between $... | {
"ground_truth": "0.30",
"style": "rule"
} | {
"index": "31554",
"split": "train"
} |
math | [
{
"content": "The integer \\( n \\) has a total of 10 divisors. These divisors are arranged in ascending order, and the 8th divisor is \\( \\frac{n}{3} \\). Find the maximum value of the integer \\( n \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "162",
"style": "rule"
} | {
"index": "31555",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\sin^2(wx) - \\sin^2(wx - \\frac{\\pi}{6})$ ($x \\in \\mathbb{R}$, $w$ is a constant and $\\frac{1}{2} < w < 1$), the graph of function $f(x)$ is symmetric about the line $x = \\pi$.\n\n(I) Find the smallest positive period of the function $f(x)$;\n\n(II) In $\\triangle... | {
"ground_truth": "\\frac{\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "31556",
"split": "train"
} |
math | [
{
"content": "Find all the integers written as $\\overline{abcd}$ in decimal representation and $\\overline{dcba}$ in base $7$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2116",
"style": "rule"
} | {
"index": "31557",
"split": "train"
} |
math | [
{
"content": "If $\\frac{1}{9}+\\frac{1}{18}=\\frac{1}{\\square}$, what is the number that replaces the $\\square$ to make the equation true? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "31558",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)= \\begin{cases} \\log_{2}x,x > 0 \\\\ x^{2}+4x+1,x\\leqslant 0\\\\ \\end{cases}$, if the real number $a$ satisfies $f(f(a))=1$, calculate the sum of all possible values of the real number $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{15}{16} - \\sqrt{5}",
"style": "rule"
} | {
"index": "31559",
"split": "train"
} |
math | [
{
"content": "Let $T_1$ be a triangle with side lengths $2011, 2012,$ and $2013$. For $n \\ge 1$, if $T_n = \\triangle ABC$ and $D, E,$ and $F$ are the points of tangency of the incircle of $\\triangle ABC$ to the sides $AB, BC$, and $AC,$ respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE,$ a... | {
"ground_truth": "\\frac{1509}{128}",
"style": "rule"
} | {
"index": "31560",
"split": "train"
} |
math | [
{
"content": "A function is given by\n$$\nf(x)=\\ln (a x+b)+x^{2} \\quad (a \\neq 0).\n$$\n(1) If the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$ is $y=x$, find the values of $a$ and $b$.\n\n(2) If $f(x) \\leqslant x^{2}+x$ always holds, find the maximum value of $ab$. Please output the final an... | {
"ground_truth": "\\frac{e}{2}",
"style": "rule"
} | {
"index": "31561",
"split": "train"
} |
math | [
{
"content": "When $\\sqrt[3]{2700}$ is simplified, the result is $a\\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ is as small as possible. What is $a+b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "103",
"style": "rule"
} | {
"index": "31562",
"split": "train"
} |
math | [
{
"content": "Given the sequence of numbers with only even digits in their decimal representation, determine the $2014^\\text{th}$ number in the sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "62048",
"style": "rule"
} | {
"index": "31563",
"split": "train"
} |
math | [
{
"content": "There are six unmarked envelopes on a table, each containing a letter for a different person. If the mail is randomly distributed among these six people, with each person getting one letter, what is the probability that exactly three people get the right letter? Please output the final answer with... | {
"ground_truth": "\\frac{1}{18}",
"style": "rule"
} | {
"index": "31564",
"split": "train"
} |
math | [
{
"content": "Given positive integers $x$ and $y$ such that $x\\neq y$ and $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{15}$, what is the smallest possible value for $x + y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "31565",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\ln (1+x)- \\frac {x(1+λx)}{1+x}$, if $f(x)\\leqslant 0$ when $x\\geqslant 0$, calculate the minimum value of $λ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {1}{2}",
"style": "rule"
} | {
"index": "31566",
"split": "train"
} |
math | [
{
"content": "A bus can hold a maximum of 38 students. What is the minimum number of buses needed to transport 411 students? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31567",
"split": "train"
} |
math | [
{
"content": "Each vertex of the parallelogram $ABCD$ lies on the same side of the plane $S$ such that the distances of the vertices $A, B$, and $C$ from the plane $S$ are 4 cm, 6 cm, and 8 cm, respectively. The area of the projection of the parallelogram onto the plane $S$, which forms the quadrilateral $A'B'C... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "31568",
"split": "train"
} |
math | [
{
"content": "Square $ABCD$ has sides of length 3. Segments $CM$ and $CN$ divide the square's area into three equal parts. How long is segment $CM$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "31569",
"split": "train"
} |
math | [
{
"content": "In the list $2, x, y, 5$, the sum of any two adjacent numbers is constant. What is the value of $x-y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "31570",
"split": "train"
} |
math | [
{
"content": "Determine the smallest positive real number \\(x\\) such that \n\\[\\lfloor x^2 \\rfloor - x \\lfloor x \\rfloor = 7.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{71}{8}",
"style": "rule"
} | {
"index": "31571",
"split": "train"
} |
math | [
{
"content": "In a trapezoid $ABCD$ with $\\angle A = \\angle B = 90^{\\circ}$, $|AB| = 5 \\text{cm}$, $|BC| = 1 \\text{cm}$, and $|AD| = 4 \\text{cm}$, point $M$ is taken on side $AB$ such that $2 \\angle BMC = \\angle AMD$. Find the ratio $|AM| : |BM|$. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "3/2",
"style": "rule"
} | {
"index": "31572",
"split": "train"
} |
math | [
{
"content": "The area of an isosceles obtuse triangle is 8, and the median drawn to one of its equal sides is $\\sqrt{37}$. Find the cosine of the angle at the vertex. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{5}",
"style": "rule"
} | {
"index": "31573",
"split": "train"
} |
math | [
{
"content": "Let $\\theta = 25^\\circ$ be an angle such that $\\tan \\theta = \\frac{1}{6}$. Compute $\\sin^6 \\theta + \\cos^6 \\theta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{12}",
"style": "rule"
} | {
"index": "31574",
"split": "train"
} |
math | [
{
"content": "What is the constant term of the expansion of $\\left(6x+\\dfrac{1}{3x}\\right)^6$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "160",
"style": "rule"
} | {
"index": "31575",
"split": "train"
} |
math | [
{
"content": "Find all integers $n$ satisfying $n \\geq 2$ and $\\dfrac{\\sigma(n)}{p(n)-1} = n$, in which $\\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "31576",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $x^{2}-4y^{2}=4$ with left and right foci $F_{1}$ and $F_{2}$. A line passing through $F_{1}$ intersects the left branch at points $A$ and $B$. If $|AB|=3$, find the perimeter of $\\triangle AF_{2}B$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "31577",
"split": "train"
} |
math | [
{
"content": "A biologist sequentially placed 150 beetles into ten jars. In each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is no less than half the number of beetles in the tenth jar. How many beetles are in the sixth jar? Please output the final ans... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "31578",
"split": "train"
} |
math | [
{
"content": "Plot points $A, B, C$ at coordinates $(0,0),(0,1)$, and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $A B$ and $B C$. Let $X_{1}$ be the area swept out when Bobby rotates $S$ counterclockwise 45 degrees about point $A$. Let $X_{2}$ be the area swept out whe... | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "31579",
"split": "train"
} |
math | [
{
"content": "Given that there are $12$ different cards, with $3$ cards each of red, yellow, green, and blue, select $3$ cards such that they cannot all be of the same color and there can be at most $1$ blue card. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "189",
"style": "rule"
} | {
"index": "31580",
"split": "train"
} |
math | [
{
"content": "If $(x+a)(x+8)=x^{2}+bx+24$ for all values of $x$, what is the value of $a+b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "31581",
"split": "train"
} |
math | [
{
"content": "A number is formed using the digits \\(1, 2, \\ldots, 9\\). Any digit can be used more than once, but adjacent digits cannot be the same. Once a pair of adjacent digits has occurred, that pair, in that order, cannot be used again. How many digits are in the largest such number? Please output the f... | {
"ground_truth": "73",
"style": "rule"
} | {
"index": "31582",
"split": "train"
} |
math | [
{
"content": "Given $x \\gt 0$, $y \\gt 0$, and $x+y=1$, find the minimum value of $\\frac{2{x}^{2}-x+1}{xy}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2}+1",
"style": "rule"
} | {
"index": "31583",
"split": "train"
} |
math | [
{
"content": "Given that $\\{a_n\\}$ is a geometric sequence and $S_n$ is the sum of the first $n$ terms, with $S_3=2$ and $S_6=6$, find the value of $a_{13}+a_{14}+a_{15}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "31584",
"split": "train"
} |
math | [
{
"content": "$D$ is a point on side $AB$ of triangle $ABC$, satisfying $AD=2$ and $DB=8$. Let $\\angle ABC=\\alpha$ and $\\angle CAB=\\beta$. <br/>$(1)$ When $CD\\perp AB$ and $\\beta =2\\alpha$, find the value of $CD$; <br/>$(2)$ If $α+\\beta=\\frac{π}{4}$, find the maximum area of triangle $ACD$. Please outp... | {
"ground_truth": "5(\\sqrt{2} - 1)",
"style": "rule"
} | {
"index": "31585",
"split": "train"
} |
math | [
{
"content": "At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two.\n\n[color=#BF0000]Rewording of the last line for clarification:[/color]\... | {
"ground_truth": "\\left\\lfloor \\frac{n}{2} \\right\\rfloor +1",
"style": "rule"
} | {
"index": "31586",
"split": "train"
} |
math | [
{
"content": "In a diagram, the grid is composed of 1x1 squares. What is the area of the shaded region if the overall width of the grid is 15 units and its height is 5 units? Some parts are shaded in the following manner: A horizontal stretch from the left edge (6 units wide) that expands 3 units upward from th... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "31587",
"split": "train"
} |
math | [
{
"content": "Given that point $P$ is a moving point on the parabola $x^{2}=2y$, and the focus is $F$. If the fixed point is $M(1,2)$, calculate the minimum value of $(|PM|+|PF|)$ when point $P$ moves on the parabola. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "31588",
"split": "train"
} |
math | [
{
"content": "Let \\(ABCD\\) be a quadrilateral with \\(BC = CD = DA = 1\\), \\(\\angle DAB = 135^\\circ\\), and \\(\\angle ABC = 75^\\circ\\). Find \\(AB\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{6}-\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "31589",
"split": "train"
} |
math | [
{
"content": "In a factory, a total of $100$ parts were produced. Among them, A produced $0$ parts, with $35$ being qualified. B produced $60$ parts, with $50$ being qualified. Let event $A$ be \"Selecting a part from the $100$ parts at random, and the part is qualified\", and event $B$ be \"Selecting a part fr... | {
"ground_truth": "\\dfrac{7}{8}",
"style": "rule"
} | {
"index": "31590",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[\n\\sum_{n = 1}^\\infty \\frac{1}{n(n + 3)}.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "31591",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $C$: $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ $(a>0, b>0)$ with the left vertex $A$, $P$ is a point on one of the asymptotes of $C$, and $Q$ is the intersection point of $AP$ and the other asymptote of $C$. If the slope of the line $AP$ is $1$ and $A$ is the trisection point of $P... | {
"ground_truth": "\\frac{\\sqrt{10}}{3}",
"style": "rule"
} | {
"index": "31592",
"split": "train"
} |
math | [
{
"content": "Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. The perimeter of $III$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "31593",
"split": "train"
} |
math | [
{
"content": "How many whole numbers between 1 and 2000 do not contain the digit 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1457",
"style": "rule"
} | {
"index": "31594",
"split": "train"
} |
math | [
{
"content": "A caterpillar is climbing a 20-meter pole. During the day, it climbs 5 meters, and during the night, it slides down 4 meters. How long will it take for the caterpillar to reach the top of the pole? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "31595",
"split": "train"
} |
math | [
{
"content": "Given that $x+\\sin y=2008$ and $x+2008 \\cos y=2007$, where $0 \\leq y \\leq \\frac{\\pi}{2}$, find the value of $x+y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2007 + \\frac\\pi 2",
"style": "rule"
} | {
"index": "31596",
"split": "train"
} |
math | [
{
"content": "Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 10x - 6y - 34$. What is $x+y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "31597",
"split": "train"
} |
math | [
{
"content": "The area of the smallest square that will contain a circle of radius 4 is what? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "31598",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=x^{4}+a x^{3}+b x^{2}+c x+d$ be a polynomial whose roots are all negative integers. If $a+b+c+d=2009$, find $d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "528",
"style": "rule"
} | {
"index": "31599",
"split": "train"
} |
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