data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given that three roots of $f(x) = x^{4} + ax^{2} + bx + c$ are $2, -3$, and $5$, what is the value of $a + b + c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "79",
"style": "rule"
} | {
"index": "33600",
"split": "train"
} |
math | [
{
"content": "Find the angle $D A C$ given that $A B = B C$ and $A C = C D$, and the lines on which points $A, B, C, D$ lie are parallel with equal distances between adjacent lines. Point $A$ is to the left of $B$, $C$ is to the left of $B$, and $D$ is to the right of $C$. Please output the final answer within ... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "33601",
"split": "train"
} |
math | [
{
"content": "Given Farmer Euclid has a field in the shape of a right triangle with legs of lengths $5$ units and $12$ units, and he leaves a small unplanted square of side length S in the corner where the legs meet at a right angle, where $S$ is $3$ units from the hypotenuse, calculate the fraction of the fiel... | {
"ground_truth": "\\frac{431}{480}",
"style": "rule"
} | {
"index": "33602",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle with side lengths $AB=6, AC=7,$ and $BC=8.$ Let $H$ be the orthocenter of $\\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC.$ $\\tfrac{[ABH']}{[ACH']}$ can be expressed as $\\frac{p}{q}$ . Find $p+q$ .\n\n*2022 CCA Math ... | {
"ground_truth": "251",
"style": "rule"
} | {
"index": "33603",
"split": "train"
} |
math | [
{
"content": "For any real number \\(x\\), \\(\\lfloor x \\rfloor\\) denotes the largest integer less than or equal to \\(x\\). For example, \\(\\lfloor 4.2 \\rfloor = 4\\) and \\(\\lfloor 0.9 \\rfloor = 0\\).\n\nIf \\(S\\) is the sum of all integers \\(k\\) with \\(1 \\leq k \\leq 999999\\) and for which \\(k\... | {
"ground_truth": "999999000",
"style": "rule"
} | {
"index": "33604",
"split": "train"
} |
math | [
{
"content": "Josef and Timothy play a game in which Josef picks an integer between 1 and 1000 inclusive and Timothy divides 1000 by that integer and states whether or not the quotient is an integer. How many integers could Josef pick such that Timothy's quotient is an integer? Please output the final answer w... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "33605",
"split": "train"
} |
math | [
{
"content": "Margo walks to her friend's house in 10 minutes. Using the same route, it takes Margo 20 minutes to get back home. If her average walking rate for the entire trip is 4 miles per hour, how many total miles did she walk? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "33606",
"split": "train"
} |
math | [
{
"content": "Three friends are in the park. Bob and Clarise are standing at the same spot and Abe is standing 10 m away. Bob chooses a random direction and walks in this direction until he is 10 m from Clarise. What is the probability that Bob is closer to Abe than Clarise is to Abe? Please output the final an... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "33607",
"split": "train"
} |
math | [
{
"content": "Find all functions $f$ from the reals to the reals such that \\[ \\left(f(x)+f(z)\\right)\\left(f(y)+f(t)\\right)=f(xy-zt)+f(xt+yz) \\] for all real $x,y,z,t$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = 0, \\quad f(x) = \\frac{1}{2}, \\quad f(x) = x^2.",
"style": "rule"
} | {
"index": "33608",
"split": "train"
} |
math | [
{
"content": "The TV station continuously plays 5 advertisements, consisting of 3 different commercial advertisements and 2 different Olympic promotional advertisements. The requirements are that the last advertisement must be an Olympic promotional advertisement, and the 2 Olympic promotional advertisements ca... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "33609",
"split": "train"
} |
math | [
{
"content": "What number results when 50 is increased by $120\\%$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "33610",
"split": "train"
} |
math | [
{
"content": "In rectangle \\(ABCD\\), points \\(E\\) and \\(F\\) lie on sides \\(AB\\) and \\(CD\\) respectively such that both \\(AF\\) and \\(CE\\) are perpendicular to diagonal \\(BD\\). Given that \\(BF\\) and \\(DE\\) separate \\(ABCD\\) into three polygons with equal area, and that \\(EF = 1\\), find the... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "33611",
"split": "train"
} |
math | [
{
"content": "The perpendicular bisectors of the sides of triangle $PQR$ meet its circumcircle at points $P',$ $Q',$ and $R',$ respectively. If the perimeter of triangle $PQR$ is 30 and the radius of the circumcircle is 7, then find the area of hexagon $PQ'RP'QR'.$ Please output the final answer within \\boxed{... | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "33612",
"split": "train"
} |
math | [
{
"content": "Given four real numbers $-9$, $a\\_1$, $a\\_2$, and $-1$ that form an arithmetic sequence, and five real numbers $-9$, $b\\_1$, $b\\_2$, $b\\_3$, and $-1$ that form a geometric sequence, find the value of $b\\_2(a\\_2-a\\_1)=$\\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "33613",
"split": "train"
} |
math | [
{
"content": "The four-corner codes for the characters \"华\", \"杯\", and \"赛\" are $2440$, $4199$, and $3088$, respectively. By concatenating these, the encoded value for \"华杯赛\" is $244041993088$. If the digits in the odd positions remain unchanged and the digits in the even positions are replaced with their c... | {
"ground_truth": "254948903981",
"style": "rule"
} | {
"index": "33614",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, there is a circle $C_{1}$: $(x-2)^{2}+(y-4)^{2}=20$. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. For $C_{2}$: $\\theta= \\frac {\\pi}{3}(\\rho\\in\\mathbb{R})$.\n\n$(1)$ Fin... | {
"ground_truth": "8+5\\sqrt {3}",
"style": "rule"
} | {
"index": "33615",
"split": "train"
} |
math | [
{
"content": "Given Angie and Carlos are seated at a round table with three other people, determine the probability that Angie and Carlos are seated directly across from each other. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "33616",
"split": "train"
} |
math | [
{
"content": "Li Fang has 4 shirts of different colors, 3 skirts of different patterns, and 2 dresses of different styles. Calculate the total number of different choices she has for the May Day celebration. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "33617",
"split": "train"
} |
math | [
{
"content": "The Wolf and the three little pigs wrote a detective story \"The Three Little Pigs-2\", and then, together with Little Red Riding Hood and her grandmother, a cookbook \"Little Red Riding Hood-2\". The publisher gave the fee for both books to the pig Naf-Naf. He took his share and handed the remain... | {
"ground_truth": "700",
"style": "rule"
} | {
"index": "33618",
"split": "train"
} |
math | [
{
"content": "The diagonals of a convex quadrilateral \\(ABCD\\) intersect at point \\(E\\). Given that \\(AB = AD\\) and \\(CA\\) is the bisector of angle \\(C\\), with \\(\\angle BAD = 140^\\circ\\) and \\(\\angle BEA = 110^\\circ\\), find angle \\(CDB\\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "33619",
"split": "train"
} |
math | [
{
"content": "Given $$x \\in \\left(- \\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$$ and $$\\sin x + \\cos x = \\frac{1}{5}$$, calculate the value of $\\tan 2x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{24}{7}",
"style": "rule"
} | {
"index": "33620",
"split": "train"
} |
math | [
{
"content": "Given that the asymptote equation of the hyperbola $y^{2}+\\frac{x^2}{m}=1$ is $y=\\pm \\frac{\\sqrt{3}}{3}x$, find the value of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "33621",
"split": "train"
} |
math | [
{
"content": "A point \\( M \\) is chosen on the diameter \\( AB \\). Points \\( C \\) and \\( D \\), lying on the circumference on one side of \\( AB \\), are chosen such that \\(\\angle AMC=\\angle BMD=30^{\\circ}\\). Find the diameter of the circle given that \\( CD=12 \\). Please output the final answer wit... | {
"ground_truth": "8\\sqrt{3}",
"style": "rule"
} | {
"index": "33622",
"split": "train"
} |
math | [
{
"content": "Consider a $5 \\times 5$ grid of squares. Vladimir colors some of these squares red, such that the centers of any four red squares do not form an axis-parallel rectangle (i.e. a rectangle whose sides are parallel to those of the squares). What is the maximum number of squares he could have colored... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "33623",
"split": "train"
} |
math | [
{
"content": "How many positive perfect square integers are factors of the product $\\left(2^{10}\\right)\\left(3^{12}\\right)\\left(5^{15}\\right)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "336",
"style": "rule"
} | {
"index": "33624",
"split": "train"
} |
math | [
{
"content": "How many numbers are in the list $ -48, -41, -34, \\ldots, 65, 72?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "33625",
"split": "train"
} |
math | [
{
"content": "Consider a decreasing arithmetic sequence $\\{a\\_n\\}$ with the sum of its first $n$ terms denoted as $S\\_n$. If $a\\_3a\\_5=63$ and $a\\_2+{a}\\_{6} =16$,\n\n(1) Find the general term formula of the sequence.\n(2) For what value of $n$ does $S\\_n$ reach its maximum value? Also, find the maximu... | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "33626",
"split": "train"
} |
math | [
{
"content": "Among the natural numbers not exceeding 10,000, calculate the number of odd numbers with distinct digits. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2605",
"style": "rule"
} | {
"index": "33627",
"split": "train"
} |
math | [
{
"content": "Find the value of $b$ such that the following equation in base $b$ is true:\n$$\\begin{array}{c@{}c@{}c@{}c@{}c@{}c@{}c}\n&&8&7&3&6&4_b\\\\\n&+&9&2&4&1&7_b\\\\\n\\cline{2-7}\n&1&8&5&8&7&1_b.\n\\end{array}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "33628",
"split": "train"
} |
math | [
{
"content": "If \\( e^{i \\theta} = \\frac{3 + i \\sqrt{2}}{4}, \\) then find \\( \\cos 3\\theta. \\) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{64}",
"style": "rule"
} | {
"index": "33629",
"split": "train"
} |
math | [
{
"content": "Circle $A$ has radius $100$. Circle $B$ has an integer radius $r<100$ and remains internally tangent to circle $A$ as it rolls once around the circumference of circle $A$. The two circles have the same points of tangency at the beginning and end of circle $B$'s trip. How many possible values can $... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "33630",
"split": "train"
} |
math | [
{
"content": "Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of\n$$\\frac{a}{b}+\\frac{b}{c}+\\frac{c}{d}+\\frac{d}{a}.$$\n[i]Israel[/i] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "33631",
"split": "train"
} |
math | [
{
"content": "Let \\( n \\) be the product of 3659893456789325678 and 342973489379256. Determine the number of digits of \\( n \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "33632",
"split": "train"
} |
math | [
{
"content": "Each of a group of $50$ girls is blonde or brunette and is blue eyed of brown eyed. If $14$ are blue-eyed blondes, $31$ are brunettes, and $18$ are brown-eyed, then the number of brown-eyed brunettes is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "33633",
"split": "train"
} |
math | [
{
"content": "Given positive integers \\(a\\), \\(b\\) \\((a \\leq b)\\), a sequence \\(\\{ f_{n} \\}\\) satisfies:\n\\[ f_{1} = a, \\, f_{2} = b, \\, f_{n+2} = f_{n+1} + f_{n} \\text{ for } n = 1, 2, \\ldots \\]\n\nIf for any positive integer \\(n\\), it holds that \n\\[ \\left( \\sum_{k=1}^{n} f_{k} \\right)^... | {
"ground_truth": "2 + \\sqrt{5}",
"style": "rule"
} | {
"index": "33634",
"split": "train"
} |
math | [
{
"content": "An amusement park has a collection of scale models, with a ratio of $1: 20$, of buildings and other sights from around the country. The height of the United States Capitol is $289$ feet. What is the height in feet of its duplicate to the nearest whole number? Please output the final answer within ... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "33635",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a\\_n\\}$ with its sum of the first $n$ terms denoted as $S\\_n$. For any $n ∈ ℕ^∗$, it is known that $S\\_n = 2(a\\_n - 1)$.\n\n1. Find the general term formula for the sequence $\\{a\\_n\\}$.\n2. Insert $k$ numbers between $a\\_k$ and $a_{k+1}$ to form an arithmetic sequence... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "33636",
"split": "train"
} |
math | [
{
"content": "In the rectangle PQRS, side PQ is of length 2 and side QR is of length 4. Points T and U lie inside the rectangle such that ∆PQT and ∆RSU are equilateral triangles. Calculate the area of quadrilateral QRUT. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4 - \\sqrt{3}",
"style": "rule"
} | {
"index": "33637",
"split": "train"
} |
math | [
{
"content": "There is a machine with 8 toys in it that each cost between 25 cents and 2 dollars, with each toy being 25 cents more expensive than the next most expensive one. Each time Sam presses the big red button on the machine, the machine randomly selects one of the remaining toys and gives Sam the option... | {
"ground_truth": "\\dfrac{6}{7}",
"style": "rule"
} | {
"index": "33638",
"split": "train"
} |
math | [
{
"content": "Given the constraints \\(x + 2y \\leq 5\\), \\(2x + y \\leq 4\\), \\(x \\geq 0\\), and \\(y \\geq 0\\), find the coordinates \\((x, y)\\) where \\(3x + 4y\\) achieves its maximum value, and determine that maximum value. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "33639",
"split": "train"
} |
math | [
{
"content": "Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of $n$? ... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "33640",
"split": "train"
} |
math | [
{
"content": "In order for Mateen to walk a kilometer(1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "33641",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin \\alpha + \\cos \\alpha = \\frac{\\sqrt{2}}{3}$, and $0 < \\alpha < \\pi$, find $\\tan\\left(\\alpha - \\frac{\\pi}{4}\\right) = \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "33642",
"split": "train"
} |
math | [
{
"content": "In the equation $\\frac {x(x - 1) - (m + 1)}{(x - 1)(m - 1)} = \\frac {x}{m}$ the roots are equal when Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "33643",
"split": "train"
} |
math | [
{
"content": "Define the ☆ operation: observe the following expressions: \n$1$☆$2=1\\times 3+2=5$; \n$4$☆$(-1)=4\\times 3-1=11$; \n$(-5)$☆$3=(-5)\\times 3+3=-12$; \n$(-6)$☆$(-3)=(-6)\\times 3-3=-21\\ldots$ \n$(1)-1$☆$2=$______, $a$☆$b=$______; \n$(2)$ If $a \\lt b$, then $a$☆$b-b$☆$a$ ______ 0 (use \"$ \\... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "33644",
"split": "train"
} |
math | [
{
"content": "In the geometric sequence $\\{a_n\\}$, if $a_5 + a_6 + a_7 + a_8 = \\frac{15}{8}$ and $a_6a_7 = -\\frac{9}{8}$, then find the value of $\\frac{1}{a_5} + \\frac{1}{a_6} + \\frac{1}{a_7} + \\frac{1}{a_8}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{5}{3}",
"style": "rule"
} | {
"index": "33645",
"split": "train"
} |
math | [
{
"content": "Brad has a cylindrical barrel with radius 10 inches and height 15 inches. He wants to fill it from a well, using a bucket in the shape of a hemisphere with a radius of 10 inches. How many trips must Brad make to the well in order to fill the barrel? Please output the final answer within \\boxed{}... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "33646",
"split": "train"
} |
math | [
{
"content": "How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "33647",
"split": "train"
} |
math | [
{
"content": "The area of the base of a hemisphere is $144\\pi$. The hemisphere is mounted on top of a cylinder that has the same radius as the hemisphere and a height of 10. What is the total surface area of the combined solid? Express your answer in terms of $\\pi$. Please output the final answer within \\box... | {
"ground_truth": "672\\pi",
"style": "rule"
} | {
"index": "33648",
"split": "train"
} |
math | [
{
"content": "For distinct real numbers $x$ and $y$, let $M(x,y)$ be the larger of $x$ and $y$ and let $m(x,y)$ be the smaller of $x$ and $y$. If $a<b<c<d<e$, then\n$M(M(a,m(b,c)),m(d,m(a,e)))=$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "b",
"style": "rule"
} | {
"index": "33649",
"split": "train"
} |
math | [
{
"content": "For positive integers $n$, let $L(n)$ be the largest factor of $n$ other than $n$ itself. Determine the number of ordered pairs of composite positive integers $(m, n)$ for which $L(m) L(n)=80$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "33650",
"split": "train"
} |
math | [
{
"content": "Calculate $(-1)^{53} + 3^{(2^3 + 5^2 - 4!)}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19682",
"style": "rule"
} | {
"index": "33651",
"split": "train"
} |
math | [
{
"content": "A tournament among 2021 ranked teams is played over 2020 rounds. In each round, two teams are selected uniformly at random among all remaining teams to play against each other. The better ranked team always wins, and the worse ranked team is eliminated. Let $p$ be the probability that the second b... | {
"ground_truth": "674",
"style": "rule"
} | {
"index": "33652",
"split": "train"
} |
math | [
{
"content": "Let \\( f(x) = x^2 + px + q \\). It is known that the inequality \\( |f(x)| > \\frac{1}{2} \\) has no solutions on the interval \\([2, 4]\\). Find \\( \\underbrace{f(f(\\ldots f}_{2017}\\left(\\frac{5-\\sqrt{11}}{2}\\right)) \\ldots) \\). Round your answer to the nearest hundredth if necessary. Pl... | {
"ground_truth": "4.16",
"style": "rule"
} | {
"index": "33653",
"split": "train"
} |
math | [
{
"content": "Let $\\omega$ be a circle, and let $ABCD$ be a quadrilateral inscribed in $\\omega$. Suppose that $BD$ and $AC$ intersect at a point $E$. The tangent to $\\omega$ at $B$ meets line $AC$ at a point $F$, so that $C$ lies between $E$ and $F$. Given that $AE=6, EC=4, BE=2$, and $BF=12$, find $DA$. Ple... | {
"ground_truth": "2 \\sqrt{42}",
"style": "rule"
} | {
"index": "33654",
"split": "train"
} |
math | [
{
"content": "Find \\[\\left|\\left(\\frac 23+\\frac 56i\\right)^8\\right|\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2825761}{1679616}",
"style": "rule"
} | {
"index": "33655",
"split": "train"
} |
math | [
{
"content": "A flagpole is originally $5$ meters tall. A hurricane snaps the flagpole at a point $x$ meters above the ground so that the upper part, still attached to the stump, touches the ground $1$ meter away from the base. What is $x$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "2.4",
"style": "rule"
} | {
"index": "33656",
"split": "train"
} |
math | [
{
"content": "Given a right prism $ABC-A_{1}B_{1}C_{1}$, where $AB=3$, $AC=4$, and $AB \\perp AC$, $AA_{1}=2$, find the sum of the surface areas of the inscribed sphere and the circumscribed sphere of the prism. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "33\\pi",
"style": "rule"
} | {
"index": "33657",
"split": "train"
} |
math | [
{
"content": "The base of a triangle is 20; the medians drawn to the lateral sides are 18 and 24. Find the area of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "288",
"style": "rule"
} | {
"index": "33658",
"split": "train"
} |
math | [
{
"content": "Determine the number of ways to arrange the letters of the word \"SUCCESS\". Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "33659",
"split": "train"
} |
math | [
{
"content": "Let $P_1,P_2,\\dots,P_n$ be $n$ distinct points over a line in the plane ($n\\geq2$). Consider all the circumferences with diameters $P_iP_j$ ($1\\leq{i,j}\\leq{n}$) and they are painted with $k$ given colors. Lets call this configuration a ($n,k$)-cloud.\n\nFor each positive integer $k$, find all... | {
"ground_truth": "n \\geq 2^k + 1",
"style": "rule"
} | {
"index": "33660",
"split": "train"
} |
math | [
{
"content": "If $g(x) = x^2$ and $f(x) = 2x - 1$, what is the value of $f(g(2))$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "33661",
"split": "train"
} |
math | [
{
"content": "Given the exponential function y=f(x) whose graph passes through the point $\\left( \\frac{1}{2}, \\frac{\\sqrt{2}}{2} \\right)$, find the value of $\\log_2 f(2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "33662",
"split": "train"
} |
math | [
{
"content": "If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "33663",
"split": "train"
} |
math | [
{
"content": "Given that a hyperbola shares common foci $F_1$ and $F_2$ with the ellipse $\\dfrac {x^{2}}{9}+ \\dfrac {y^{2}}{25}=1$, and their sum of eccentricities is $2 \\dfrac {4}{5}$. \n$(1)$ Find the standard equation of the hyperbola; \n$(2)$ Let $P$ be a point of intersection between the hyperbola and... | {
"ground_truth": "- \\dfrac {1}{7}",
"style": "rule"
} | {
"index": "33664",
"split": "train"
} |
math | [
{
"content": "Let $AB$ be a diameter of a circle centered at $O$. Let $E$ be a point on the circle, and let the tangent at $B$ intersect the tangent at $E$ and $AE$ at $C$ and $D$, respectively. If $\\angle BAE = 43^\\circ$, find $\\angle CED$, in degrees.\n\n[asy]\nimport graph;\n\nunitsize(2 cm);\n\npair O,... | {
"ground_truth": "47^\\circ",
"style": "rule"
} | {
"index": "33665",
"split": "train"
} |
math | [
{
"content": "Calculate $(-1)^{45} + 2^{(3^2+5^2-4^2)}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "262143",
"style": "rule"
} | {
"index": "33666",
"split": "train"
} |
math | [
{
"content": "Given Elmer’s new car provides a 70% better fuel efficiency and uses a type of fuel that is 35% more expensive per liter, calculate the percentage by which Elmer will save money on fuel costs if he uses his new car for his journey. Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "20.6\\%",
"style": "rule"
} | {
"index": "33667",
"split": "train"
} |
math | [
{
"content": "Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that\n\\[P(1) = P(3) = P(5) = P(7) = a\\]and\n\\[P(2) = P(4) = P(6) = P(8) = -a.\\]What is the smallest possible value of $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": " 315",
"style": "rule"
} | {
"index": "33668",
"split": "train"
} |
math | [
{
"content": "Given the parabola $y^2 = 4x$, a line passing through point $P(4, 0)$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. Find the minimum value of $y_1^2 + y_2^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "33669",
"split": "train"
} |
math | [
{
"content": "An angle $x$ is chosen at random from the interval $0^\\circ < x < 90^\\circ.$ Let $p$ be the probability that the numbers $\\sin^2 x, \\cos^2 x,$ and $\\sin x \\cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n,$ where $d$ is the number of degrees in $\\text{arctan}$ $m$ ... | {
"ground_truth": "92",
"style": "rule"
} | {
"index": "33670",
"split": "train"
} |
math | [
{
"content": "Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ of $\\triangle ABC$, respectively, and $\\overrightarrow{m}=(a,−\\sqrt {3}b)$, $\\overrightarrow{n}=(\\sin B,\\cos A)$, if $a= \\sqrt {7}$, $b=2$, and $\\overrightarrow{m} \\perp \\overrightarrow{n}$, then the ar... | {
"ground_truth": "\\frac{3\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "33671",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b} = -\\sqrt{3}$, calculate the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Please output the final answer... | {
"ground_truth": "\\frac{5\\pi}{6}",
"style": "rule"
} | {
"index": "33672",
"split": "train"
} |
math | [
{
"content": "For natural numbers, when $P$ is divided by $D$, the quotient is $Q$ and the remainder is $R$. When $Q$ is divided by $D'$, the quotient is $Q'$ and the remainder is $R'$. Then, when $P$ is divided by $DD'$, the remainder is: Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "$R+R'D$",
"style": "rule"
} | {
"index": "33673",
"split": "train"
} |
math | [
{
"content": "Given a random variable $\\xi$ that follows the normal distribution $N(0, \\sigma^2)$, if $P(\\xi > 2) = 0.023$, calculate $P(-2 \\leq \\xi \\leq 2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.954",
"style": "rule"
} | {
"index": "33674",
"split": "train"
} |
math | [
{
"content": "A journey is divided into three segments: uphill, flat, and downhill. The ratio of the lengths of these segments is $1: 2: 3$, and the ratio of the times taken to walk each segment is $4: 5: 6$. Given that the person’s speed uphill is 3 km/h and the total distance is 50 km, how much time does the ... | {
"ground_truth": "10.4166666667",
"style": "rule"
} | {
"index": "33675",
"split": "train"
} |
math | [
{
"content": "If $2x+7$ is a factor of $6x^3+19x^2+cx+35$, find $c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "33676",
"split": "train"
} |
math | [
{
"content": "If $x$ is an integer, find the largest integer that always divides the expression \\[(12x + 2)(8x + 14)(10x + 10)\\] when $x$ is odd. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "33677",
"split": "train"
} |
math | [
{
"content": "For a positive integer \\( n \\), let \\( s(n) \\) denote the sum of its digits, and let \\( p(n) \\) denote the product of its digits. If the equation \\( s(n) + p(n) = n \\) holds true, then \\( n \\) is called a coincidence number. What is the sum of all coincidence numbers? Please output the f... | {
"ground_truth": "531",
"style": "rule"
} | {
"index": "33678",
"split": "train"
} |
math | [
{
"content": "While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\\lfloor 1000a\\rfloor$ . Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "941",
"style": "rule"
} | {
"index": "33679",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\\sqrt{3}b\\cos A=a\\cos B$.\n\n$(1)$ Find the angle $A$;\n\n$(2)$ If $a= \\sqrt{2}$ and $\\frac{c}{a}= \\frac{\\sin A}{\\sin B}$, find the perimeter of $\\trian... | {
"ground_truth": "3 \\sqrt{2}",
"style": "rule"
} | {
"index": "33680",
"split": "train"
} |
math | [
{
"content": "A chemistry student conducted an experiment: starting with a bottle filled with syrup solution, the student poured out one liter of liquid, refilled the bottle with water, then poured out one liter of liquid again, and refilled the bottle with water once more. As a result, the syrup concentration ... | {
"ground_truth": "1.2",
"style": "rule"
} | {
"index": "33681",
"split": "train"
} |
math | [
{
"content": "Compute $\\frac{x^{10} - 32x^5 + 1024}{x^5 - 32}$ when $x=8$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32768",
"style": "rule"
} | {
"index": "33682",
"split": "train"
} |
math | [
{
"content": "The graph of the function $f(x)=\\sqrt{3}\\cos 2x-\\sin 2x$ can be obtained by translating the graph of the function $f(x)=2\\sin 2x$ by an unspecified distance. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{\\pi}{6}",
"style": "rule"
} | {
"index": "33683",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)= \\begin{cases} a+\\ln x,x > 0 \\\\ g(x)-x,x < 0\\\\ \\end{cases}$, which is an odd function, and $g(-e)=0$, find the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1-e",
"style": "rule"
} | {
"index": "33684",
"split": "train"
} |
math | [
{
"content": "A polyhedron has 12 faces and is such that:\n(i) all faces are isosceles triangles,\n(ii) all edges have length either \\( x \\) or \\( y \\),\n(iii) at each vertex either 3 or 6 edges meet, and\n(iv) all dihedral angles are equal.\nFind the ratio \\( x / y \\). Please output the final answer with... | {
"ground_truth": "3/5",
"style": "rule"
} | {
"index": "33685",
"split": "train"
} |
math | [
{
"content": "The overall idea is a common method in mathematical problem-solving. Below is the train of thought for factoring the polynomial $(a^{2}+2a)(a^{2}+2a+2)+1$: Consider \"$a^{2}+2a$\" as a whole, let $a^{2}+2a=x$, then the expression $=x(x+2)+1=x^{2}+2x+1=(x+1)^{2}$, then restore \"$x$\" to \"$a^{2}+2... | {
"ground_truth": "2024",
"style": "rule"
} | {
"index": "33686",
"split": "train"
} |
math | [
{
"content": "One fine summer day, François was looking for Béatrice in Cabourg. Where could she be? Perhaps on the beach (one chance in two) or on the tennis court (one chance in four), it could be that she is in the cafe (also one chance in four). If Béatrice is on the beach, which is large and crowded, Franç... | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "33687",
"split": "train"
} |
math | [
{
"content": "A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "33688",
"split": "train"
} |
math | [
{
"content": "2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{14}{39}",
"style": "rule"
} | {
"index": "33689",
"split": "train"
} |
math | [
{
"content": "The two squares shown share the same center $O_{}$ and have sides of length 1. The length of $\\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$\n[asy] //code taken from thread for problem real alpha =... | {
"ground_truth": "185",
"style": "rule"
} | {
"index": "33690",
"split": "train"
} |
math | [
{
"content": "Given that the equation about $x$, $x^{2}-2a\\ln x-2ax=0$ has a unique solution, find the value of the real number $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "33691",
"split": "train"
} |
math | [
{
"content": "On the coordinate plane, the points \\(A(0, 2)\\), \\(B(1, 7)\\), \\(C(10, 7)\\), and \\(D(7, 1)\\) are given. Find the area of the pentagon \\(A B C D E\\), where \\(E\\) is the intersection point of the lines \\(A C\\) and \\(B D\\). Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "33692",
"split": "train"
} |
math | [
{
"content": "Let $A B C D$ be a square of side length 10 . Point $E$ is on ray $\\overrightarrow{A B}$ such that $A E=17$, and point $F$ is on ray $\\overrightarrow{A D}$ such that $A F=14$. The line through $B$ parallel to $C E$ and the line through $D$ parallel to $C F$ meet at $P$. Compute the area of quadr... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "33693",
"split": "train"
} |
math | [
{
"content": "In the repeating decimal 0.2017, if the sum of all digits from the $m$-th digit to the $n$-th digit after the decimal point is 2017, find the value of $n$ when $m$ takes the minimum value. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "808",
"style": "rule"
} | {
"index": "33694",
"split": "train"
} |
math | [
{
"content": "The area of each of the four congruent L-shaped regions of this 100-inch by 100-inch square is 3/16 of the total area. How many inches long is the side of the center square? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "33695",
"split": "train"
} |
math | [
{
"content": "Arc $AC$ is a quarter-circle with center $B$. The shaded region $ABC$ is \"rolled\" along a straight board $PQ$ until it reaches its original orientation for the first time with point $B$ landing at point $B^{\\prime}$. If $BC = \\frac{2}{\\pi}$ cm, what is the length of the path that point $B$ tr... | {
"ground_truth": "3\\text{ cm}",
"style": "rule"
} | {
"index": "33696",
"split": "train"
} |
math | [
{
"content": "The function \\(f(x) = 5x^2 - 15x - 2\\) has a minimum value when x is negative. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-13.25",
"style": "rule"
} | {
"index": "33697",
"split": "train"
} |
math | [
{
"content": "Given the numbers from $000$ to $999$, calculate how many have three digits in non-decreasing or non-increasing order, including cases where digits can repeat. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "430",
"style": "rule"
} | {
"index": "33698",
"split": "train"
} |
math | [
{
"content": "In triangle $PQR$, $PQ = 8$, $QR = 15$, and $PR = 17$. Point $S$ is the angle bisector of $\\angle QPR$. Find the length of $QS$ and then find the length of the altitude from $P$ to $QS$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "33699",
"split": "train"
} |
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