data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In this diagram, not drawn to scale, Figures $I$ and $III$ are equilateral triangular regions with respective areas of $32\\sqrt{3}$ and $8\\sqrt{3}$ square inches. Figure $II$ is a square region with area $32$ square inches. Let the length of segment $AD$ be decreased by $12\\frac{1}{2}$% of itse... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "4000",
"split": "train"
} |
math | [
{
"content": "How many prime numbers are between 20 and 30? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "4001",
"split": "train"
} |
math | [
{
"content": "Given cos(π/4 - α) = 3/5 and sin(5π/4 + β) = -12/13, where α ∈ (π/4, 3π/4) and β ∈ (0, π/4), find tan(α)/tan(β). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-17",
"style": "rule"
} | {
"index": "4002",
"split": "train"
} |
math | [
{
"content": "If $128^3 = 16^x$, what is the value of $2^{-x}$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2^{5.25}}",
"style": "rule"
} | {
"index": "4003",
"split": "train"
} |
math | [
{
"content": "We define \\( a \\star b = a \\times a - b \\times b \\). Find the value of \\( 3 \\star 2 + 4 \\star 3 + 5 \\star 4 + \\cdots + 20 \\star 19 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "396",
"style": "rule"
} | {
"index": "4004",
"split": "train"
} |
math | [
{
"content": "Let $a_1, a_2, \\ldots$ and $b_1, b_2, \\ldots$ be arithmetic progressions such that $a_1 = 25, b_1 = 75$, and $a_{100} + b_{100} = 100$. Find the sum of the first hundred terms of the progression $a_1 + b_1, a_2 + b_2, \\ldots$ Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "10,000",
"style": "rule"
} | {
"index": "4005",
"split": "train"
} |
math | [
{
"content": "Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of $0$, and the score increases by $1$ for each like vote and decreases by $1$ for each dislike vote. At one point Sangho saw that his video had a score of $90$, and that $... | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "4006",
"split": "train"
} |
math | [
{
"content": "What is the number of ways in which one can choose $60$ unit squares from a $11 \\times 11$ chessboard such that no two chosen squares have a side in common? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "62",
"style": "rule"
} | {
"index": "4007",
"split": "train"
} |
math | [
{
"content": "Let $a_{1}, a_{2}, \\ldots, a_{2005}$ be real numbers such that $$\\begin{array}{ccccccccccc} a_{1} \\cdot 1 & + & a_{2} \\cdot 2 & + & a_{3} \\cdot 3 & + & \\cdots & + & a_{2005} \\cdot 2005 & = & 0 \\\\ a_{1} \\cdot 1^{2} & + & a_{2} \\cdot 2^{2} & + & a_{3} \\cdot 3^{2} & + & \\cdots & + & a_{2... | {
"ground_truth": "1 / 2004!",
"style": "rule"
} | {
"index": "4008",
"split": "train"
} |
math | [
{
"content": "For the largest natural \\( m \\), when will the product \\( m! \\cdot 2022! \\) be a factorial of a natural number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2022! - 1",
"style": "rule"
} | {
"index": "4009",
"split": "train"
} |
math | [
{
"content": "A brand of orange juice is available in shop $A$ and shop $B$ at an original price of $\\$2.00$ per bottle. Shop $A$ provides a \"buy 4 get 1 free\" promotion and shop $B$ provides a $15\\%$ discount if one buys 4 bottles or more. Find the minimum cost (in cents) if one wants to buy 13 bottles of ... | {
"ground_truth": "2160",
"style": "rule"
} | {
"index": "4010",
"split": "train"
} |
math | [
{
"content": "The distance from point P(1, -1) to the line $ax+3y+2a-6=0$ is maximized when the line passing through P is perpendicular to the given line. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{2}",
"style": "rule"
} | {
"index": "4011",
"split": "train"
} |
math | [
{
"content": "The point \\( N \\) is the center of the face \\( ABCD \\) of the cube \\( ABCDEFGH \\). Also, \\( M \\) is the midpoint of the edge \\( AE \\). If the area of \\(\\triangle MNH\\) is \\( 13 \\sqrt{14} \\), what is the edge length of the cube? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "2\\sqrt{13}",
"style": "rule"
} | {
"index": "4012",
"split": "train"
} |
math | [
{
"content": "Rectangle $EFGH$ has area $2016$. An ellipse with area $2016\\pi$ passes through $E$ and $G$ and has foci at $F$ and $H$. What is the perimeter of the rectangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8\\sqrt{1008}",
"style": "rule"
} | {
"index": "4013",
"split": "train"
} |
math | [
{
"content": "A point M on the parabola $y=-4x^2$ is at a distance of 1 from the focus. What is the y-coordinate of point M? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{15}{16}",
"style": "rule"
} | {
"index": "4014",
"split": "train"
} |
math | [
{
"content": "In a bag, there are $5$ balls of the same size, including $3$ red balls and $2$ white balls.<br/>$(1)$ If one ball is drawn with replacement each time, and this process is repeated $3$ times, with the number of times a red ball is drawn denoted as $X$, find the probability distribution and expecta... | {
"ground_truth": "\\frac{108}{625}",
"style": "rule"
} | {
"index": "4015",
"split": "train"
} |
math | [
{
"content": "For the real numbers \\(a\\) and \\(b\\), it holds that \\(a^{2} + 4b^{2} = 4\\). How large can \\(3a^{5}b - 40a^{3}b^{3} + 48ab^{5}\\) be? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "4016",
"split": "train"
} |
math | [
{
"content": "A certain high school is planning to hold a coming-of-age ceremony for senior students on the \"May Fourth\" Youth Day to motivate the seniors who are preparing for the college entrance examination. The Student Affairs Office has prepared five inspirational songs, a video speech by an outstanding ... | {
"ground_truth": "20160",
"style": "rule"
} | {
"index": "4017",
"split": "train"
} |
math | [
{
"content": "Given that $E$ and $F$ are a chord of the circle $C: x^{2}+y^{2}-2x-4y+1=0$, and $CE\\bot CF$, $P$ is the midpoint of $EF$. When the chord $EF$ moves on the circle $C$, there exist two points $A$ and $B$ on the line $l: x-y-3=0$ such that $∠APB≥\\frac{π}{2}$ always holds. Determine the minimum val... | {
"ground_truth": "4\\sqrt{2}",
"style": "rule"
} | {
"index": "4018",
"split": "train"
} |
math | [
{
"content": "The vertices of a quadrilateral lie on the graph of $y=\\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\\ln{\\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex? Please output the final answer within \\boxed{}.... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "4019",
"split": "train"
} |
math | [
{
"content": "Given that the left focus of the ellipse $$\\frac {x^{2}}{a^{2}}$$+ $$\\frac {y^{2}}{b^{2}}$$\\=1 (a＞b＞0) is F, the left vertex is A, and the upper vertex is B. If the distance from point F to line AB is $$\\frac {2b}{ \\sqrt {17}}$$, then the eccentricity of the ellipse is \\_\\_\\_\\_\\_\\_. Ple... | {
"ground_truth": "\\frac {1}{3}",
"style": "rule"
} | {
"index": "4020",
"split": "train"
} |
math | [
{
"content": "Over all real numbers $x$ and $y$, find the minimum possible value of $$ (x y)^{2}+(x+7)^{2}+(2 y+7)^{2} $$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "4021",
"split": "train"
} |
math | [
{
"content": "Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9\\pi}{2} - 9",
"style": "rule"
} | {
"index": "4022",
"split": "train"
} |
math | [
{
"content": "If 5 times a number is 2, then 100 times the reciprocal of the number is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "4023",
"split": "train"
} |
math | [
{
"content": "Solve the following system of equations:\n$$x+\\frac{1}{x^3}=2y,\\quad y+\\frac{1}{y^3}=2z,\\quad z+\\frac{1}{z^3}=2w,\\quad w+\\frac{1}{w^3}=2x.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(1, 1, 1, 1) \\text{ and } (-1, -1, -1, -1)",
"style": "rule"
} | {
"index": "4024",
"split": "train"
} |
math | [
{
"content": "Quadrilateral $ALEX,$ pictured below (but not necessarily to scale!)\ncan be inscribed in a circle; with $\\angle LAX = 20^{\\circ}$ and $\\angle AXE = 100^{\\circ}:$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "4025",
"split": "train"
} |
math | [
{
"content": "Yasmine makes her own chocolate beverage by mixing volumes of milk and syrup in the ratio \\(5: 2\\). Milk comes in \\(2 \\text{ L}\\) bottles and syrup comes in \\(1.4 \\text{ L}\\) bottles. Yasmine has a limitless supply of full bottles of milk and of syrup. Determine the smallest volume of choc... | {
"ground_truth": "19.6",
"style": "rule"
} | {
"index": "4026",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[\\sin^2 6^\\circ + \\sin^2 12^\\circ + \\sin^2 18^\\circ + \\dots + \\sin^2 174^\\circ.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15.5",
"style": "rule"
} | {
"index": "4027",
"split": "train"
} |
math | [
{
"content": "Solve\n\\[\\arcsin (\\sin x) = \\frac{x}{2}.\\]Enter all the solutions, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2 \\pi}{3}, 0, \\frac{2 \\pi}{3}",
"style": "rule"
} | {
"index": "4028",
"split": "train"
} |
math | [
{
"content": "Given $f(\\alpha) = \\frac{\\sin(\\pi - \\alpha)\\cos(\\pi + \\alpha)\\sin(-\\alpha + \\frac{3\\pi}{2})}{\\cos(-\\alpha)\\cos(\\alpha + \\frac{\\pi}{2})}$.\n\n$(1)$ Simplify $f(\\alpha)$; \n\n$(2)$ If $\\alpha$ is an angle in the third quadrant, and $\\cos(\\alpha - \\frac{3\\pi}{2}) = \\frac{1}{... | {
"ground_truth": "\\frac{2\\sqrt{6}}{5}",
"style": "rule"
} | {
"index": "4029",
"split": "train"
} |
math | [
{
"content": "$ABCD$ is a rectangular sheet of paper. $E$ and $F$ are points on $AB$ and $CD$ respectively such that $BE < CF$. If $BCFE$ is folded over $EF$, $C$ maps to $C'$ on $AD$ and $B$ maps to $B'$ such that $\\angle{AB'C'} \\cong \\angle{B'EA}$. If $AB' = 5$ and $BE = 23$, then the area of $ABCD$ can be... | {
"ground_truth": "338",
"style": "rule"
} | {
"index": "4030",
"split": "train"
} |
math | [
{
"content": "Let the function \\( f(x) \\) be defined on \\([0,1]\\), satisfying \\( f(0) = f(1) \\) and for any \\( x, y \\in [0,1] \\) having \\( |f(x) - f(y)| < |x - y| \\). Find the smallest real number \\( m \\) such that for any function \\( f(x) \\) meeting the above conditions and for any \\( x, y \\in... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "4031",
"split": "train"
} |
math | [
{
"content": "For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$ . Find the minimum possible value for $|p(15)|$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "47",
"style": "rule"
} | {
"index": "4032",
"split": "train"
} |
math | [
{
"content": "A line passes through the point $(-2, \\sqrt{3})$ at an angle of inclination of $\\frac{\\pi}{3}$. Find the parametric equation of the line, and if it intersects the curve $y^2 = -x - 1$ at points A and B, find the length $|AB|$. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "\\frac{10}{3}",
"style": "rule"
} | {
"index": "4033",
"split": "train"
} |
math | [
{
"content": "Find the value of the following expressions without using a calculator:\n\\\\((1)\\\\lg 5^{2}+ \\frac {2}{3}\\lg 8+\\lg 5\\lg 20+(\\lg 2)^{2}\\\\)\n\\\\((2)\\\\) Let \\(2^{a}=5^{b}=m\\), and \\(\\frac {1}{a}+ \\frac {1}{b}=2\\), find \\(m\\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\sqrt {10}",
"style": "rule"
} | {
"index": "4034",
"split": "train"
} |
math | [
{
"content": "Given a right triangle \\( \\triangle ABC \\) with legs \\( AC=12 \\), \\( BC=5 \\). Point \\( D \\) is a moving point on the hypotenuse \\( AB \\). The triangle is folded along \\( CD \\) to form a right dihedral angle \\( A-CD-B \\). When the length of \\( AB \\) is minimized, let the plane angl... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "4035",
"split": "train"
} |
math | [
{
"content": "Find the value of $\\frac{5x+9y}{45xy}$, given $x = \\frac{3}{5}$ and $y = \\frac{7}{9}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10}{21}",
"style": "rule"
} | {
"index": "4036",
"split": "train"
} |
math | [
{
"content": "Add $175_{9} + 714_{9} + 61_9$. Express your answer in base $9$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1061_{9}",
"style": "rule"
} | {
"index": "4037",
"split": "train"
} |
math | [
{
"content": "$\\triangle ABC$ and $\\triangle DBC$ share $BC$. $AB = 5\\ \\text{cm}$, $AC = 12\\ \\text{cm}$, $DC = 8\\ \\text{cm}$, and $BD = 20\\ \\text{cm}$. What is the least possible integral number of centimeters in $BC$?\n\n[asy]\nsize(100); import graph; currentpen = fontsize(10pt);\npair B = (0,0), C ... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "4038",
"split": "train"
} |
math | [
{
"content": "Carla rotated point $A$ 420 degrees clockwise about point $B$ and it landed at point $C$. Devon rotated the original point $A$ $x$ degrees counterclockwise about point $B$ and it also landed at point $C$. If $x<360$, what is the value of $x$? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "300^{\\circ}",
"style": "rule"
} | {
"index": "4039",
"split": "train"
} |
math | [
{
"content": "What is the largest value of $n$ less than 100,000 for which the expression $8(n-2)^5-n^2+14n-24$ is a multiple of 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "99997",
"style": "rule"
} | {
"index": "4040",
"split": "train"
} |
math | [
{
"content": "In the \"Triangle\" cinema, the seats are arranged in the shape of a triangle: the first row has one seat numbered 1, the second row has seats numbered 2 and 3, the third row has seats numbered 4, 5, 6, and so on. We define the best seat in the cinema hall as the one located at the center of the h... | {
"ground_truth": "1035",
"style": "rule"
} | {
"index": "4041",
"split": "train"
} |
math | [
{
"content": "Find the inverse of the matrix\n\\[\\begin{pmatrix} 2 & 3 \\\\ -1 & 7 \\end{pmatrix}.\\]If the inverse does not exist, then enter the zero matrix. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} 7/17 & -3/17 \\\\ 1/17 & 2/17 \\end{pmatrix}",
"style": "rule"
} | {
"index": "4042",
"split": "train"
} |
math | [
{
"content": "In the quadrilateral \\(ABCD\\), \\(AB = 2\\), \\(BC = 4\\), \\(CD = 5\\). Find its area given that it is both circumscribed and inscribed. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{30}",
"style": "rule"
} | {
"index": "4043",
"split": "train"
} |
math | [
{
"content": "The hyperbola \\[\\frac{(x-3)^2}{5^2} - \\frac{(y-17)^2}{12^2} = 1\\]has two foci, which have different $x-$coordinates. Find the coordinates of the one with the larger $x-$coordinate. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(16,17)",
"style": "rule"
} | {
"index": "4044",
"split": "train"
} |
math | [
{
"content": "Given real numbers $x$, $y$ satisfying $x > y > 0$, and $x + y \\leqslant 2$, the minimum value of $\\dfrac{2}{x+3y}+\\dfrac{1}{x-y}$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac {3+2 \\sqrt {2}}{4}",
"style": "rule"
} | {
"index": "4045",
"split": "train"
} |
math | [
{
"content": "Compute $\\sqrt[4]{12960000}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "4046",
"split": "train"
} |
math | [
{
"content": "Find the largest real number \\( p \\) such that all three roots of the equation below are positive integers:\n\\[ \n5x^{3} - 5(p+1)x^{2} + (71p-1)x + 1 = 66p .\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "76",
"style": "rule"
} | {
"index": "4047",
"split": "train"
} |
math | [
{
"content": "A grocer creates a display of cans where the top row contains two cans and each subsequent lower row has three more cans than the row preceding it. If the total number of cans used in the display is 120, how many rows are there in the display? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "n = 9",
"style": "rule"
} | {
"index": "4048",
"split": "train"
} |
math | [
{
"content": "Let $P(x)$ be a polynomial of degree at most 3 such that $P(x)=\\frac{1}{1+x+x^{2}}$ for $x=1,2,3,4$. What is $P(5) ?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{-3}{91}",
"style": "rule"
} | {
"index": "4049",
"split": "train"
} |
math | [
{
"content": "In the coordinate plane, a rectangle has vertices with coordinates $(34,0), (41,0), (34,9), (41,9)$. Find the smallest value of the parameter $a$ such that the line $y = ax$ divides this rectangle into two parts where the area of one part is twice the area of the other. If the answer is not an int... | {
"ground_truth": "0.08",
"style": "rule"
} | {
"index": "4050",
"split": "train"
} |
math | [
{
"content": "Determine the value of $a^3 + b^3$ given that $a+b=12$ and $ab=20$, and also return the result for $(a+b-c)(a^3+b^3)$, where $c=a-b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4032",
"style": "rule"
} | {
"index": "4051",
"split": "train"
} |
math | [
{
"content": "Barry picks infinitely many points inside a unit circle, each independently and uniformly at random, $P_{1}, P_{2}, \\ldots$ Compute the expected value of $N$, where $N$ is the smallest integer such that $P_{N+1}$ is inside the convex hull formed by the points $P_{1}, P_{2}, \\ldots, P_{N}$. Submi... | {
"ground_truth": "6.54",
"style": "rule"
} | {
"index": "4052",
"split": "train"
} |
math | [
{
"content": "Is there a triangle with $12 \\, cm^2$ area and $12$ cm perimeter? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\text{No}",
"style": "rule"
} | {
"index": "4053",
"split": "train"
} |
math | [
{
"content": "A prime number of the form \\( 2^{p} - 1 \\) is called a Mersenne prime, where \\( p \\) is also a prime number. To date, the largest known Mersenne prime is \\( 2^{82589933} - 1 \\). Find the last two digits of this number. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "91",
"style": "rule"
} | {
"index": "4054",
"split": "train"
} |
math | [
{
"content": "Which of the following expressions is equal to an odd integer for every integer $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2017+2n",
"style": "rule"
} | {
"index": "4055",
"split": "train"
} |
math | [
{
"content": "Juan rolls a fair regular octahedral die marked with the numbers 1 through 8. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "4056",
"split": "train"
} |
math | [
{
"content": "Given an angle $α$ whose terminal side passes through the point $(-1, \\sqrt{2})$, find the values of $\\tan α$ and $\\cos 2α$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{3}",
"style": "rule"
} | {
"index": "4057",
"split": "train"
} |
math | [
{
"content": "In triangle \\( PQR \\), the median \\( PA \\) and the angle bisector \\( QB \\) (where \\( A \\) and \\( B \\) are the points of their intersection with the corresponding sides of the triangle) intersect at point \\( O \\). It is known that \\( 3PQ = 5QR \\). Find the ratio of the area of triangl... | {
"ground_truth": "2.6",
"style": "rule"
} | {
"index": "4058",
"split": "train"
} |
math | [
{
"content": "Square $ABCD$ is constructed along diameter $AB$ of a semicircle, as shown. The semicircle and square $ABCD$ are coplanar. Line segment $AB$ has a length of 6 centimeters. If point $M$ is the midpoint of arc $AB$, what is the length of segment $MC$? Express your answer in simplest radical form. [a... | {
"ground_truth": "3\\sqrt{10}",
"style": "rule"
} | {
"index": "4059",
"split": "train"
} |
math | [
{
"content": "The coefficient of $x^5$ in the expansion of $(1+x)^2(1-x)^5$ is \\_\\_\\_\\_\\_\\_ (Answer with a number). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "4060",
"split": "train"
} |
math | [
{
"content": "For how many integers $x$ is the point $(x, -x)$ inside or on the circle of radius $10$ centered at $(5, 5)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "4061",
"split": "train"
} |
math | [
{
"content": "Let $ABCDEF$ be a regular hexagon, and let $G,H,I$ be the midpoints of sides $AB,CD,EF$ respectively. If the area of $\\triangle GHI$ is $225$, what is the area of hexagon $ABCDEF$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "4062",
"split": "train"
} |
math | [
{
"content": "The hypotenuse of a right triangle measures 10 inches and one angle is $45^{\\circ}$. What is the number of square inches in the area of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "4063",
"split": "train"
} |
math | [
{
"content": "Let $P$ be a point inside triangle $ABC$. Let $G_1$, $G_2$, and $G_3$ be the centroids of triangles $PBC$, $PCA$, and $PAB$, respectively. If the area of triangle $ABC$ is 18, then find the area of triangle $G_1 G_2 G_3$.\n\n[asy]\nimport geometry;\n\nunitsize(2 cm);\n\npair A, B, C, P;\npair[] ... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "4064",
"split": "train"
} |
math | [
{
"content": "Let $S=1+ \\frac{1}{ \\sqrt {2}}+ \\frac{1}{ \\sqrt {3}}+...+ \\frac{1}{ \\sqrt {1000000}}$. Determine the integer part of $S$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1998",
"style": "rule"
} | {
"index": "4065",
"split": "train"
} |
math | [
{
"content": "If $\\alpha \\in (0, \\frac{\\pi}{2})$, and $\\tan 2\\alpha = \\frac{\\cos \\alpha}{2-\\sin \\alpha}$, calculate the value of $\\tan \\alpha$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{15}}{15}",
"style": "rule"
} | {
"index": "4066",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\\angle APB = \\angle BPC = \\angle CPA$. Find $PC$.\n[asy] unitsize(0.2 cm); pair A, B, C, P; A = (0,14); B = (0,0); C = (21*sqrt(3),0); P = intersectionpoint(arc(B,6,0,180),arc(C,33,0,180)); draw(... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "4067",
"split": "train"
} |
math | [
{
"content": "There are numbers from 1 to 2013 on the blackboard. Each time, two numbers can be erased and replaced with the sum of their digits. This process continues until there are four numbers left, whose product is 27. What is the sum of these four numbers? Please output the final answer within \\boxed{}.... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "4068",
"split": "train"
} |
math | [
{
"content": "Let $x$ and $y$ be positive real numbers. Find the minimum value of\n\\[\\left( x + \\frac{1}{y} \\right) \\left( x + \\frac{1}{y} - 2018 \\right) + \\left( y + \\frac{1}{x} \\right) \\left( y + \\frac{1}{x} - 2018 \\right).\\] Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "-2036162",
"style": "rule"
} | {
"index": "4069",
"split": "train"
} |
math | [
{
"content": "Simplify $\\frac{1}{1+\\sqrt{3}} \\cdot \\frac{1}{1-\\sqrt{5}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{1 - \\sqrt{5} + \\sqrt{3} - \\sqrt{15}}",
"style": "rule"
} | {
"index": "4070",
"split": "train"
} |
math | [
{
"content": "Find all integers $A$ if it is known that $A^{6}$ is an eight-digit number composed of the digits $0, 1, 2, 2, 2, 3, 4, 4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "4071",
"split": "train"
} |
math | [
{
"content": "Given the numbers \\(1, 2, 3, \\ldots, 1000\\). Find the largest number \\(m\\) with the following property: no matter which \\(m\\) of these numbers are removed, among the remaining \\(1000-m\\) numbers, there will be two such that one of them divides the other. Please output the final answer wit... | {
"ground_truth": "499",
"style": "rule"
} | {
"index": "4072",
"split": "train"
} |
math | [
{
"content": "The science club has 25 members: 10 boys and 15 girls. A 5-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{475}{506}",
"style": "rule"
} | {
"index": "4073",
"split": "train"
} |
math | [
{
"content": "A math teacher requires Noelle to do one homework assignment for each of the first five homework points she wants to earn; for each of the next five homework points, she needs to do two homework assignments; and so on, so that to earn the $n^{\\text{th}}$ homework point, she has to do $n\\div5$ (r... | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "4074",
"split": "train"
} |
math | [
{
"content": "Compute $\\begin{pmatrix} 2 & 3 \\\\ 7 & -1 \\end{pmatrix} \\begin{pmatrix} 1 & -5 \\\\ 0 & 4 \\end{pmatrix}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} 2 & 2 \\\\ 7 & -39 \\end{pmatrix}",
"style": "rule"
} | {
"index": "4075",
"split": "train"
} |
math | [
{
"content": "If $w$, $x$, $y$, and $z$ are real numbers satisfying: \\begin{align*}\nw+x+y &= -2, \\\\\nw+x+z &= 4, \\\\\nw+y+z &= 19, \\text{ and} \\\\\nx+y+z &= 12,\n\\end{align*} what is $wx + yz$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "99",
"style": "rule"
} | {
"index": "4076",
"split": "train"
} |
math | [
{
"content": "Find the largest value of $x$ that satisfies the equation $|x-5|=12$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "4077",
"split": "train"
} |
math | [
{
"content": "In the diagram, $l\\|k$. What is the number of degrees in $\\angle SRQ$? [asy]\ndraw((-.4,-.4)--(2,2)--(2,-.4));\ndraw((-.5,0)--(3,0),Arrows);\ndraw((-.5,1)--(3,1),Arrows);\ndraw((1.9,0)--(1.9,.1)--(2,.1));\nlabel(\"$S$\",(1,1),NNW);\nlabel(\"$R$\",(2,2),N);\nlabel(\"$Q$\",(2,1),NE);\nlabel(\"$l$\... | {
"ground_truth": "40^\\circ",
"style": "rule"
} | {
"index": "4078",
"split": "train"
} |
math | [
{
"content": "Let $\\omega = -\\frac{1}{2} + \\frac{i\\sqrt{3}}{2}$ and $\\omega^2 = -\\frac{1}{2} - \\frac{i\\sqrt{3}}{2}$. Define $T$ as the set of all points in the complex plane of the form $a + b\\omega + c\\omega^2 + d$, where $0 \\leq a, b, c \\leq 1$ and $d \\in \\{0, 1\\}$. Find the area of $T$. Please... | {
"ground_truth": "3\\sqrt{3}",
"style": "rule"
} | {
"index": "4079",
"split": "train"
} |
math | [
{
"content": "Given a square \\(ABCD\\) with side length 1, determine the maximum value of \\(PA \\cdot PB \\cdot PC \\cdot PD\\) where point \\(P\\) lies inside or on the boundary of the square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{16}",
"style": "rule"
} | {
"index": "4080",
"split": "train"
} |
math | [
{
"content": "Find the positive integer $N$, such that numbers $N$ and $N^2$ end in the same sequence of four digits $abcd$ where $a$ is not zero, under the modulus $8000$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "625",
"style": "rule"
} | {
"index": "4081",
"split": "train"
} |
math | [
{
"content": "Shift the graph of the function $f(x)=2\\sin(2x+\\frac{\\pi}{6})$ to the left by $\\frac{\\pi}{12}$ units, and then shift it upwards by 1 unit to obtain the graph of $g(x)$. If $g(x_1)g(x_2)=9$, and $x_1, x_2 \\in [-2\\pi, 2\\pi]$, then find the maximum value of $2x_1-x_2$. Please output the final... | {
"ground_truth": "\\frac {49\\pi}{12}",
"style": "rule"
} | {
"index": "4082",
"split": "train"
} |
math | [
{
"content": "An infinite geometric series has common ratio $\\frac{-1}{3}$ and sum $25.$ What is the second term of the sequence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{-100}{9}",
"style": "rule"
} | {
"index": "4083",
"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": "15",
"style": "rule"
} | {
"index": "4084",
"split": "train"
} |
math | [
{
"content": "Kadin makes a snowman by stacking snowballs of radius 2 inches, 3 inches, and 5 inches. Assuming all his snowballs are spherical, what is the total volume of snow he uses, in cubic inches? Express your answer in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "\\frac{640}{3}\\pi",
"style": "rule"
} | {
"index": "4085",
"split": "train"
} |
math | [
{
"content": "Given that $17^{-1} \\equiv 31 \\pmod{53}$, find $36^{-1} \\pmod{53}$ as a residue modulo 53 (i.e., a value between 0 and 52 inclusive). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "4086",
"split": "train"
} |
math | [
{
"content": "Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$. Let $\\frac{m}{n}$ be the probability that $\\sqrt{2+\\sqrt{3}}\\le\\left|v+w\\right|$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "582",
"style": "rule"
} | {
"index": "4087",
"split": "train"
} |
math | [
{
"content": "What is the value of $(-2)^{3}-(-3)^{2}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-17",
"style": "rule"
} | {
"index": "4088",
"split": "train"
} |
math | [
{
"content": "What is the maximum number of numbers that can be selected from the set \\( 1, 2, \\ldots, 1963 \\) such that the sum of no two numbers is divisible by their difference? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "655",
"style": "rule"
} | {
"index": "4089",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $n$ that satisfies the following conditions: For $n$, there exists a positive integer $k$ such that $\\frac{8}{15} < \\frac{n}{n+k} < \\frac{7}{13}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "4090",
"split": "train"
} |
math | [
{
"content": "Let $Q$ be a $(2n+1) \\times (2n+1)$ board. Some of its cells are colored black in such a way that every $2 \\times 2$ board of $Q$ has at most $2$ black cells. Find the maximum amount of black cells that the board may have. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "(2n+1)(n+1)",
"style": "rule"
} | {
"index": "4091",
"split": "train"
} |
math | [
{
"content": "A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? Please output the final answer withi... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "4092",
"split": "train"
} |
math | [
{
"content": "Given positive real numbers $a$, $b$, $c$, $d$ satisfying $a^{2}-ab+1=0$ and $c^{2}+d^{2}=1$, find the value of $ab$ when $\\left(a-c\\right)^{2}+\\left(b-d\\right)^{2}$ reaches its minimum. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{2} + 1",
"style": "rule"
} | {
"index": "4093",
"split": "train"
} |
math | [
{
"content": "Let $Y$ be as in problem 14. Find the maximum $Z$ such that three circles of radius $\\sqrt{Z}$ can simultaneously fit inside an equilateral triangle of area $Y$ without overlapping each other. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10 \\sqrt{3}-15",
"style": "rule"
} | {
"index": "4094",
"split": "train"
} |
math | [
{
"content": "If \\( a \\) is the smallest cubic number divisible by 810, find the value of \\( a \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "729000",
"style": "rule"
} | {
"index": "4095",
"split": "train"
} |
math | [
{
"content": "The points \\( K, L, M, N \\) are the centers of the circles inscribed in the faces \\( S A B, S A C, S B C, \\) and \\( A B C \\) of the tetrahedron \\( S A B C \\). It is known that \\( A B = S C = 5 \\), \\( A C = S B = 7 \\), \\( B C = S A = 8 \\). Find the volume of the tetrahedron \\( K L M ... | {
"ground_truth": "0.66",
"style": "rule"
} | {
"index": "4096",
"split": "train"
} |
math | [
{
"content": "In an arcade game, the \"monster\" is the shaded sector of a circle of radius $1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $60^\\circ$. What is the perimeter of the monster in cm? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{3}\\pi + 2",
"style": "rule"
} | {
"index": "4097",
"split": "train"
} |
math | [
{
"content": "Find constants $A$, $B$, and $C$, such that\n$$\\frac{-x^2+3x-4}{x^3+x}= \\frac{A}{x} +\\frac{Bx+C}{x^2+1} $$Enter your answer as the ordered triplet $(A,B,C)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-4,3,3)",
"style": "rule"
} | {
"index": "4098",
"split": "train"
} |
math | [
{
"content": "Given integers $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $\\ldots $ satisfy the following conditions: $a_{1}=0$, $a_{2}=-|a+1|$, $a_{3}=-|a_{2}+2|$, $a_{4}=-|a_{3}+3|$, and so on, then the value of $a_{2022}$ is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1011",
"style": "rule"
} | {
"index": "4099",
"split": "train"
} |
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