data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$ . Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 + \\frac{3\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "31800",
"split": "train"
} |
math | [
{
"content": "If the domains of the functions \\( f(x) \\) and \\( g(x) \\) are both the set of non-negative real numbers, and for any \\( x \\geq 0 \\), \\( f(x) \\cdot g(x) = \\min \\{ f(x), g(x) \\} \\), given \\( f(x) = 3 - x \\) and \\( g(x) = \\sqrt{2x + 5} \\), then the maximum value of \\( f(x) \\cdot g... | {
"ground_truth": "2\\sqrt{3} - 1",
"style": "rule"
} | {
"index": "31801",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle ABC$ be an equilateral triangle with side length $s$. Point $P$ is an internal point of the triangle such that $AP=1$, $BP=\\sqrt{3}$, and $CP=2$. Find the value of $s$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\sqrt{7}$",
"style": "rule"
} | {
"index": "31802",
"split": "train"
} |
math | [
{
"content": "Given that \\(\\alpha\\) and \\(\\beta\\) are acute angles, and the following equations hold:\n$$\n\\left\\{\\begin{array}{l}\n3 \\sin ^{2} \\alpha + 2 \\sin ^{2} \\beta = 1, \\\\\n3 \\sin 2 \\alpha - 2 \\sin 2 \\beta = 0.\n\\end{array}\\right.\n$$\n\nDetermine \\(\\alpha + 2\\beta\\). Please outp... | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "31803",
"split": "train"
} |
math | [
{
"content": "Given five letters a, b, c, d, and e arranged in a row, find the number of arrangements where both a and b are not adjacent to c. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "31804",
"split": "train"
} |
math | [
{
"content": "Ten elves are sitting around a circular table, each with a basket of nuts. Each elf is asked, \"How many nuts do your two neighbors have together?\" and the answers, going around the circle, are 110, 120, 130, 140, 150, 160, 170, 180, 190, and 200. How many nuts does the elf who answered 160 have?... | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "31805",
"split": "train"
} |
math | [
{
"content": "Given a set $\\{4,6,8,12,14,18\\}$, select three different numbers, add two of these numbers, multiply their sum by the third number, and finally subtract the smallest number you initially selected. Find the smallest result that can be obtained from this process. Please output the final answer wit... | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "31806",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\cos 2x+2 \\sqrt {3}\\sin x\\cos x$,\n(1) Find the range of the function $f(x)$ and write out the interval where the function $f(x)$ is strictly increasing;\n(2) If $0 < θ < \\dfrac {π}{6}$ and $f(θ)= \\dfrac {4}{3}$, compute the value of $\\cos 2θ$. Please output the fin... | {
"ground_truth": "\\dfrac { \\sqrt {15}+2}{6}",
"style": "rule"
} | {
"index": "31807",
"split": "train"
} |
math | [
{
"content": "Inside square $ABCD$ with side $s$, quarter-circle arcs with radii $s$ and centers at $A$ and $B$ are drawn. These arcs intersect at a point $X$ inside the square. How far is $X$ from the side of $CD$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2} s(2-\\sqrt{3})",
"style": "rule"
} | {
"index": "31808",
"split": "train"
} |
math | [
{
"content": "Given that $\\overrightarrow{e}$ is a unit vector, $|\\overrightarrow{a}|=4$, and the angle between $\\overrightarrow{a}$ and $\\overrightarrow{e}$ is $\\frac{2}{3}π$, find the projection of $\\overrightarrow{a}$ onto $\\overrightarrow{e}$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "31809",
"split": "train"
} |
math | [
{
"content": "The terms $140, a, \\frac{45}{28}$ are the first, second and third terms, respectively, of a geometric sequence. If $a$ is positive, what is the value of $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "31810",
"split": "train"
} |
math | [
{
"content": "Two circles with radii of 4 and 5 are externally tangent to each other and are both circumscribed by a third circle. Find the area of the shaded region outside these two smaller circles but within the larger circle. Express your answer in terms of $\\pi$. Assume the configuration of tangency and c... | {
"ground_truth": "40\\pi",
"style": "rule"
} | {
"index": "31811",
"split": "train"
} |
math | [
{
"content": "In a table consisting of $n$ rows and $m$ columns, numbers are written such that the sum of the elements in each row is 1248, and the sum of the elements in each column is 2184. Find the numbers $n$ and $m$ for which the expression $2n - 3m$ takes the smallest possible natural value. In the answer... | {
"ground_truth": "143",
"style": "rule"
} | {
"index": "31812",
"split": "train"
} |
math | [
{
"content": "The number $2027$ is prime. Let $T = \\sum \\limits_{k=0}^{72} \\binom{2024}{k}$. What is the remainder when $T$ is divided by $2027$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1369",
"style": "rule"
} | {
"index": "31813",
"split": "train"
} |
math | [
{
"content": "At a family outing to a theme park, the Thomas family, comprising three generations, plans to purchase tickets. The two youngest members, categorized as children, get a 40% discount. The two oldest members, recognized as seniors, enjoy a 30% discount. The middle generation no longer enjoys any dis... | {
"ground_truth": "49.27",
"style": "rule"
} | {
"index": "31814",
"split": "train"
} |
math | [
{
"content": "Determine the sum of all real numbers \\(x\\) satisfying\n\\[\n(x^2 - 5x + 3)^{x^2 - 6x + 3} = 1.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "31815",
"split": "train"
} |
math | [
{
"content": "A right triangle \\(ABC\\) has a perimeter of 54, with the leg \\(AC\\) being greater than 10. A circle with a radius of 6, whose center lies on leg \\(BC\\), is tangent to lines \\(AB\\) and \\(AC\\). Find the area of triangle \\(ABC\\). Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "243/2",
"style": "rule"
} | {
"index": "31816",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. The polar equation of curve C is $\\rho - 2\\cos\\theta - 6\\sin\\theta + \\frac{1}{\\rho} = 0$, and the parametric equation of line... | {
"ground_truth": "2\\sqrt{6}",
"style": "rule"
} | {
"index": "31817",
"split": "train"
} |
math | [
{
"content": "Aria and Bianca walk at different, but constant speeds. They each begin at 8:00 a.m. from the opposite ends of a road and walk directly toward the other's starting point. They pass each other at 8:42 a.m. Aria arrives at Bianca's starting point at 9:10 a.m. When does Bianca arrive at Aria's starti... | {
"ground_truth": "9:45 a.m.",
"style": "rule"
} | {
"index": "31818",
"split": "train"
} |
math | [
{
"content": "Given $a > 0$, $b > 0$, and it satisfies the equation $3a + b = a^2 + ab$. Find the minimum value of $2a + b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2} + 3",
"style": "rule"
} | {
"index": "31819",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of $$ \\int^1_0|f'(x)|^2|f(x)|\\frac1{\\sqrt x}dx $$ over all continuously differentiable functions $f:[0,1]\\to\\mathbb R$ with $f(0)=0$ and $$ \\int^1_0|f'(x)|^2dx\\le1. $$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "31820",
"split": "train"
} |
math | [
{
"content": "In three-dimensional space, find the number of lattice points that have a distance of 3 from the origin.\n\nNote: A point is a lattice point if all its coordinates are integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "31821",
"split": "train"
} |
math | [
{
"content": "In the following diagram, $AB=50$. Find $AX$.\n\n[asy]\nimport markers;\n\nreal t=.67;\npair A=(0,0);\npair B=(3,-2);\npair C=(1.5,1.5);\npair X=t*A+(1-t)*B;\n\ndraw(C--A--B--C--X);\n\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,E);\nlabel(\"$C$\",C,N);\nlabel(\"$X$\",X,SW);\n\nmarkangle(n=1,radius=15,... | {
"ground_truth": "\\frac{50}3",
"style": "rule"
} | {
"index": "31822",
"split": "train"
} |
math | [
{
"content": "Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "31823",
"split": "train"
} |
math | [
{
"content": "We denote by $\\mathbb{R}^\\plus{}$ the set of all positive real numbers.\n\nFind all functions $f: \\mathbb R^ \\plus{} \\rightarrow\\mathbb R^ \\plus{}$ which have the property:\n\\[f(x)f(y)\\equal{}2f(x\\plus{}yf(x))\\]\nfor all positive real numbers $x$ and $y$.\n\n[i] Please output the final ... | {
"ground_truth": "f(x) = 2",
"style": "rule"
} | {
"index": "31824",
"split": "train"
} |
math | [
{
"content": "Evaluate the polynomial \\[ p(x) = x^4 - 3x^3 - 9x^2 + 27x - 8, \\] where $x$ is a positive number such that $x^2 - 3x - 9 = 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{65 + 81\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "31825",
"split": "train"
} |
math | [
{
"content": "For how many ordered pairs of positive integers $(x,y),$ with $y<x\\le 100,$ are both $\\frac xy$ and $\\frac{x+1}{y+1}$ integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "31826",
"split": "train"
} |
math | [
{
"content": "A merchant bought some goods at a discount of $20\\%$ of the list price. He wants to mark them at such a price that he can give a discount of $20\\%$ of the marked price and still make a profit of $20\\%$ of the selling price. The per cent of the list price at which he should mark them is: Please ... | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "31827",
"split": "train"
} |
math | [
{
"content": "The line joining $(2,3)$ and $(5,1)$ divides the square shown into two parts. What fraction of the area of the square is above this line? The square has vertices at $(2,1)$, $(5,1)$, $(5,4)$, and $(2,4)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "31828",
"split": "train"
} |
math | [
{
"content": "Let \\( n = 2^{31} \\times 3^{19} \\times 5^7 \\). How many positive integer divisors of \\( n^2 \\) are less than \\( n \\) but do not divide \\( n \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13307",
"style": "rule"
} | {
"index": "31829",
"split": "train"
} |
math | [
{
"content": "The vertices $V$ of a centrally symmetric hexagon in the complex plane are given by \\[V=\\left\\{ \\sqrt{2}i,-\\sqrt{2}i, \\frac{1}{\\sqrt{8}}(1+i),\\frac{1}{\\sqrt{8}}(-1+i),\\frac{1}{\\sqrt{8}}(1-i),\\frac{1}{\\sqrt{8}}(-1-i) \\right\\}.\\]For each $j$, $1\\leq j\\leq 12$, an element $z_j$ is... | {
"ground_truth": "233",
"style": "rule"
} | {
"index": "31830",
"split": "train"
} |
math | [
{
"content": "Jeff decides to play with a Magic 8 Ball. Each time he asks it a question, it has a 1/3 chance of giving him a positive answer. If he asks it 7 questions, what is the probability that it gives him exactly 3 positive answers? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "\\frac{560}{2187}",
"style": "rule"
} | {
"index": "31831",
"split": "train"
} |
math | [
{
"content": "Simplify $\\cos 20^\\circ - \\cos 40^\\circ.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5} - 1}{4}",
"style": "rule"
} | {
"index": "31832",
"split": "train"
} |
math | [
{
"content": "There is a rectangle $ABCD$ such that $AB=12$ and $BC=7$ . $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that $\\frac{AE}{EB} = 1$ and $\\frac{CF}{FD} = \\frac{1}{2}$ . Call $X$ the intersection of $AF$ and $DE$ . What is the area of pentagon $BCFXE$ ?\n\n\nPropose... | {
"ground_truth": "47",
"style": "rule"
} | {
"index": "31833",
"split": "train"
} |
math | [
{
"content": "The remainders when three positive integers are divided by 5 are 1, 2, and 3. Find the remainder when their product is divided by 5. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "31834",
"split": "train"
} |
math | [
{
"content": "If the function \\( f(x) = (x^2 - 1)(x^2 + ax + b) \\) satisfies \\( f(x) = f(4 - x) \\) for any \\( x \\in \\mathbb{R} \\), what is the minimum value of \\( f(x) \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-16",
"style": "rule"
} | {
"index": "31835",
"split": "train"
} |
math | [
{
"content": "Equilateral $\\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside the plane of $\\triangle ABC$ and are on opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\\triangle PAB$ and $\\triangle QAB$ form a $120^{\\circ}$ dihedral angle (the angle ... | {
"ground_truth": "450",
"style": "rule"
} | {
"index": "31836",
"split": "train"
} |
math | [
{
"content": "One day Max says to Liz, \"Out of the 25 people taking either English or French, you and I are the only two taking both.'' Liz, being mathematically inclined, responds by pointing out that there are exactly twice as many people in the English class as there are in the French class. How many people... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "31837",
"split": "train"
} |
math | [
{
"content": "Calculate $3 \\cdot 7^{-1} + 6 \\cdot 13^{-1} \\pmod{60}$.\n\nExpress your answer as an integer from $0$ to $59$, inclusive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "31838",
"split": "train"
} |
math | [
{
"content": "Point \\( O \\) is located inside an isosceles right triangle \\( ABC \\). The distance from \\( O \\) to vertex \\( A \\) (the right angle) is 6, to vertex \\( B \\) is 9, and to vertex \\( C \\) is 3. Find the area of triangle \\( ABC \\). Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "\\frac{45}{2} + 9\\sqrt{2}",
"style": "rule"
} | {
"index": "31839",
"split": "train"
} |
math | [
{
"content": "Express $0.3\\overline{45}$ as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{83}{110}",
"style": "rule"
} | {
"index": "31840",
"split": "train"
} |
math | [
{
"content": "Two motorcyclists departed simultaneously from points \\( A \\) and \\( B \\) towards each other and met 50 km from point \\( B \\). After arriving at points \\( A \\) and \\( B \\), the motorcyclists immediately turned back and met for the second time 25 km from point \\( A \\). How many kilomete... | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "31841",
"split": "train"
} |
math | [
{
"content": "Let $A$ be the set of all real numbers $a$ that satisfy $\\left(a-2\\right)x^{2}+2\\left(a-2\\right)x-4 \\lt 0$ for any $x\\in R$. Let $B$ be the set of all real numbers $x$ that satisfy $\\left(a-2\\right)x^{2}+2\\left(a-2\\right)x-4 \\lt 0$ for any $a\\in \\left[-2,2\\right]$. Find $A\\cap (\\co... | {
"ground_truth": "\\{-1\\}",
"style": "rule"
} | {
"index": "31842",
"split": "train"
} |
math | [
{
"content": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl}\n-x + 3 & \\text{if } x \\le 0, \\\\\n2x - 5 & \\text{if } x > 0.\n\\end{array}\n\\right.\\]How many solutions does the equation $f(f(x)) = 4$ have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "31843",
"split": "train"
} |
math | [
{
"content": "A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?\n[asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); d... | {
"ground_truth": "\\frac{\\sqrt{2} - 1}{2}",
"style": "rule"
} | {
"index": "31844",
"split": "train"
} |
math | [
{
"content": "What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "31845",
"split": "train"
} |
math | [
{
"content": "Given a parallelogram \\(A B C D\\) with \\(\\angle B = 111^\\circ\\) and \\(B C = B D\\). On the segment \\(B C\\), there is a point \\(H\\) such that \\(\\angle B H D = 90^\\circ\\). Point \\(M\\) is the midpoint of side \\(A B\\). Find the angle \\(A M H\\). Provide the answer in degrees. Pleas... | {
"ground_truth": "132",
"style": "rule"
} | {
"index": "31846",
"split": "train"
} |
math | [
{
"content": "Calculate the sum of the integers 122 and 78, express both numbers and the resulting sum in base-5. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1300_5",
"style": "rule"
} | {
"index": "31847",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)= \\sqrt{3}\\sin \\omega x+\\cos (\\omega x+ \\frac{\\pi}{3})+\\cos (\\omega x- \\frac{\\pi}{3})-1$ ($\\omega > 0$, $x\\in\\mathbb{R}$),\n\nand the smallest positive period of the function $f(x)$ is $\\pi$.\n\n$(1)$ Find the analytical expression of the function $f(x)$;\n\n... | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"index": "31848",
"split": "train"
} |
math | [
{
"content": "A cube with an edge length of 2 Chinese feet is cut 4 times horizontally and then 5 times vertically. What is the total surface area of all the small blocks after cutting, in square Chinese feet? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "31849",
"split": "train"
} |
math | [
{
"content": "Find how many positive integers $k\\in\\{1,2,\\ldots,2022\\}$ have the following property: if $2022$ real numbers are written on a circle so that the sum of any $k$ consecutive numbers is equal to $2022$ then all of the $2022$ numbers are equal. Please output the final answer within \\bo... | {
"ground_truth": "672",
"style": "rule"
} | {
"index": "31850",
"split": "train"
} |
math | [
{
"content": "Four male students and five female students are lined up in a row. Calculate the number of different arrangements with alternating male and female students. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2880",
"style": "rule"
} | {
"index": "31851",
"split": "train"
} |
math | [
{
"content": "Addison's age is three times Brenda's age. Janet is six years older than Brenda. Addison and Janet are twins. How old is Brenda? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "31852",
"split": "train"
} |
math | [
{
"content": "The graphs of the lines $y = \\frac{3}{2}x$ and $y=2x - 1$ intersect and form an acute angle. Find the slope $k$ of the angle bisector of this acute angle that is oriented positively with respect to the x-axis. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7 - \\sqrt{29}}{4}",
"style": "rule"
} | {
"index": "31853",
"split": "train"
} |
math | [
{
"content": "Let $T = \\{ 1, 2, 3, \\dots, 14, 15 \\}$ . Say that a subset $S$ of $T$ is *handy* if the sum of all the elements of $S$ is a multiple of $5$ . For example, the empty set is handy (because its sum is 0) and $T$ itself is handy (because its sum is 120). Compute the number of handy sub... | {
"ground_truth": "6560",
"style": "rule"
} | {
"index": "31854",
"split": "train"
} |
math | [
{
"content": "From a circular piece of paper with radius $BC$, Jeff removes the unshaded sector shown. Using the larger shaded sector, he joins edge $BC$ to edge $BA$ (without overlap) to form a cone of radius 12 centimeters and of volume $432\\pi$ cubic centimeters. What is the number of degrees in the measu... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "31855",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\\overrightarrow{m} = (\\cos B, \\cos C)$ and $\\overrightarrow{n} = (2a + c, b)$, and $\\overrightarrow{m} \\perp \\overrightarrow{n}$.\n\n(I) Find the measure of angle $B$ and the... | {
"ground_truth": "\\frac{3\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "31856",
"split": "train"
} |
math | [
{
"content": "The digits 1, 2, 3, 4 and 9 are each used once to form the smallest possible even five-digit number. The digit in the tens place is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "31857",
"split": "train"
} |
math | [
{
"content": "Mary and Sally were once the same height. Since then, Sally grew \\( 20\\% \\) taller and Mary's height increased by half as many centimetres as Sally's height increased. Sally is now 180 cm tall. How tall, in cm, is Mary now? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "165",
"style": "rule"
} | {
"index": "31858",
"split": "train"
} |
math | [
{
"content": "Determine all pairs $(a, b)$ of integers with the property that the numbers $a^{2}+4 b$ and $b^{2}+4 a$ are both perfect squares. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-4,-4),(-5,-6),(-6,-5),(0, k^{2}),(k^{2}, 0),(k, 1-k)",
"style": "rule"
} | {
"index": "31859",
"split": "train"
} |
math | [
{
"content": "For $\\mathbf{v} = \\begin{pmatrix} 1 \\\\ y \\end{pmatrix}$ and $\\mathbf{w} = \\begin{pmatrix} 9 \\\\ 3 \\end{pmatrix}$,\n\\[\\text{proj}_{\\mathbf{w}} \\mathbf{v} = \\begin{pmatrix} -6 \\\\ -2 \\end{pmatrix}.\\]Find $y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "-23",
"style": "rule"
} | {
"index": "31860",
"split": "train"
} |
math | [
{
"content": "Given the set of digits {1, 2, 3, 4, 5}, find the number of three-digit numbers that can be formed with the digits 2 and 3, where 2 is positioned before 3. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "31861",
"split": "train"
} |
math | [
{
"content": "Let \\( f(x) = mx^2 + (2n + 1)x - m - 2 \\) (where \\( m, n \\in \\mathbb{R} \\) and \\( m \\neq 0 \\)) have at least one root in the interval \\([3, 4]\\). Find the minimum value of \\( m^2 + n^2 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.01",
"style": "rule"
} | {
"index": "31862",
"split": "train"
} |
math | [
{
"content": "What is the smallest base-10 integer that can be represented as $AA_5$ and $BB_7$, where $A$ and $B$ are valid digits in their respective bases? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24_{10}",
"style": "rule"
} | {
"index": "31863",
"split": "train"
} |
math | [
{
"content": "Given a cube $ABCDEFGH$ with an edge length of $6$, let $M$ and $N$ be points on $BB_1$ and $B_1C_1$ respectively, such that $B_1M = B_1N = 2$. Let $S$ and $P$ be the midpoints of segments $AD$ and $MN$ respectively. Find the distance between the skew lines $SP$ and $AC_1$. Please output the final... | {
"ground_truth": "\\frac{\\sqrt{114}}{38}",
"style": "rule"
} | {
"index": "31864",
"split": "train"
} |
math | [
{
"content": "Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\\cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. What is the smallest possible value of $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "492",
"style": "rule"
} | {
"index": "31865",
"split": "train"
} |
math | [
{
"content": "Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by 2 and add 1 or he may multiply the stored value by 4 and add 3. The first player to make the stored value exceed $2^{100}$ wins. The stored value starts at 1 and Mark goes first. ... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "31866",
"split": "train"
} |
math | [
{
"content": "The sultan gathered 300 court sages and proposed a trial. There are 25 different colors of hats, known in advance to the sages. The sultan informed them that each sage would be given one of these hats, and if they wrote down the number of hats for each color, all these numbers would be different. ... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "31867",
"split": "train"
} |
math | [
{
"content": "What fraction of the area of an isosceles trapezoid $KLMN (KL \\parallel MN)$ is the area of triangle $ABC$, where $A$ is the midpoint of base $KL$, $B$ is the midpoint of base $MN$, and $C$ is the midpoint of leg $KN$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "31868",
"split": "train"
} |
math | [
{
"content": "For a finite sequence $B = (b_1, b_2, \\dots, b_{50})$ of numbers, the Cesaro sum of $B$ is defined as\n\\[\\frac{T_1 + \\cdots + T_{50}}{50},\\]\nwhere $T_k = b_1 + \\cdots + b_k$ for $1 \\leq k \\leq 50$.\n\nIf the Cesaro sum of the 50-term sequence $(b_1, \\dots, b_{50})$ is 200, what is the Ce... | {
"ground_truth": "198.078431372549",
"style": "rule"
} | {
"index": "31869",
"split": "train"
} |
math | [
{
"content": "Given an ellipse with its foci on the x-axis and its lower vertex at D(0, -1), the eccentricity of the ellipse is $e = \\frac{\\sqrt{6}}{3}$. A line L passes through the point P(0, 2).\n(Ⅰ) Find the standard equation of the ellipse.\n(Ⅱ) If line L is tangent to the ellipse, find the equation of li... | {
"ground_truth": "\\frac{3\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "31870",
"split": "train"
} |
math | [
{
"content": "Given \\( a, b > 0 \\). Find the maximum values of \\( \\sqrt{\\frac{a}{2a+b}} + \\sqrt{\\frac{b}{2b+a}} \\) and \\( \\sqrt{\\frac{a}{a+2b}} + \\sqrt{\\frac{b}{b+2a}} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "31871",
"split": "train"
} |
math | [
{
"content": "Given plane vectors $\\vec{a}, \\vec{b}, \\vec{c}$ satisfying $|\\vec{a}| = |\\vec{b}|$, $\\vec{c} = \\lambda \\vec{a} + \\mu \\vec{b}$, $|\\vec{c}| = 1 + \\vec{a} \\cdot \\vec{b}$, and $(\\vec{a} + \\vec{b}) \\cdot \\vec{c} = 1$, find the minimum possible value of $\\frac{|\\vec{a} - \\vec{c}|}{|... | {
"ground_truth": "2 - \\sqrt{2}",
"style": "rule"
} | {
"index": "31872",
"split": "train"
} |
math | [
{
"content": "A teen age boy wrote his own age after his father's. From this new four place number, he subtracted the absolute value of the difference of their ages to get $4,289$ . The sum of their ages was Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "59",
"style": "rule"
} | {
"index": "31873",
"split": "train"
} |
math | [
{
"content": "An equilateral triangle and a regular hexagon have equal perimeters. If the triangle's area is 4, what is the area of the hexagon? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "31874",
"split": "train"
} |
math | [
{
"content": "Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is $a... | {
"ground_truth": "152",
"style": "rule"
} | {
"index": "31875",
"split": "train"
} |
math | [
{
"content": "A regular hexagon has an area of $150\\sqrt{3}$ cm². If each side of the hexagon is decreased by 3 cm, by how many square centimeters is the area decreased? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "76.5\\sqrt{3}",
"style": "rule"
} | {
"index": "31876",
"split": "train"
} |
math | [
{
"content": "Let $x, y, z$ be real numbers such that $x + y + z = 2$, and $x \\ge -\\frac{2}{3}$, $y \\ge -1$, and $z \\ge -2$. Find the maximum value of \n\\[\\sqrt{3x + 2} + \\sqrt{3y + 4} + \\sqrt{3z + 7}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{57}",
"style": "rule"
} | {
"index": "31877",
"split": "train"
} |
math | [
{
"content": "Given a cubic polynomial $ 60 x^3 - 80x^2 + 24x - 2 $, which has three roots $\\alpha, \\beta, \\gamma$, all distinct and lying in the interval $(0, 1)$. Find the sum \n\\[\n\\frac{1}{1-\\alpha} + \\frac{1}{1-\\beta} + \\frac{1}{1-\\gamma}.\n\\] Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "31878",
"split": "train"
} |
math | [
{
"content": "Among all triangles $ABC,$ find the maximum value of $\\cos A + \\cos B \\cos C.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "31879",
"split": "train"
} |
math | [
{
"content": "For positive numbers $x$ and $y$ the operation $\\spadesuit (x,y)$ is defined as \n\\[\\spadesuit (x,y) = x-\\dfrac{1}{y}\\]\nWhat is $\\spadesuit (2,\\spadesuit (2,2))$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "31880",
"split": "train"
} |
math | [
{
"content": "$g(x):\\mathbb{Z}\\rightarrow\\mathbb{Z}$ is a function that satisfies $$ g(x)+g(y)=g(x+y)-xy. $$ If $g(23)=0$ , what is the sum of all possible values of $g(35)$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "210",
"style": "rule"
} | {
"index": "31881",
"split": "train"
} |
math | [
{
"content": "For what smallest natural $n$ is each of the fractions\n\n$$\n\\frac{7}{n+9}, \\frac{8}{n+10}, \\ldots, \\frac{31}{n+33}\n$$\n\nin its simplest form? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "31882",
"split": "train"
} |
math | [
{
"content": "Given the fraction \\(\\frac{5}{1+\\sqrt[3]{32 \\cos ^{4} 15^{\\circ}-10-8 \\sqrt{3}}}\\). Simplify the expression under the cubic root to a simpler form, and then reduce the fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 - \\sqrt[3]{4} + \\sqrt[3]{16}",
"style": "rule"
} | {
"index": "31883",
"split": "train"
} |
math | [
{
"content": "Given circle $C$: $x^{2}+y^{2}+8x+ay-5=0$ passes through the focus of parabola $E$: $x^{2}=4y$. The length of the chord formed by the intersection of the directrix of parabola $E$ and circle $C$ is $\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4 \\sqrt{6}",
"style": "rule"
} | {
"index": "31884",
"split": "train"
} |
math | [
{
"content": "Consider a $2 \\times 2$ grid of squares. Each of the squares will be colored with one of 10 colors, and two colorings are considered equivalent if one can be rotated to form the other. How many distinct colorings are there? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "2530",
"style": "rule"
} | {
"index": "31885",
"split": "train"
} |
math | [
{
"content": "A total of $960$ people are randomly numbered from $1$ to $960$. Using systematic sampling, $32$ people are selected for a survey. Find the number of people to be selected from those with numbers falling within $[450,750]$. Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31886",
"split": "train"
} |
math | [
{
"content": "In a tournament, there are 16 chess players. Determine the number of different possible schedules for the first round (schedules are considered different if they differ by the participants of at least one match; the color of the pieces and the board number are not considered). Please output the fi... | {
"ground_truth": "2027025",
"style": "rule"
} | {
"index": "31887",
"split": "train"
} |
math | [
{
"content": "How many values of $x,-19<x<98$, satisfy $$\\cos ^{2} x+2 \\sin ^{2} x=1 ?$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "31888",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=(a+ \\frac {1}{a})\\ln x-x+ \\frac {1}{x}$, where $a > 0$. \n(I) If $f(x)$ has an extreme value point in $(0,+\\infty)$, find the range of values for $a$; \n(II) Let $a\\in(1,e]$, when $x_{1}\\in(0,1)$, $x_{2}\\in(1,+\\infty)$, denote the maximum value of $f(x_{2})-f(x_{... | {
"ground_truth": "\\frac {4}{e}",
"style": "rule"
} | {
"index": "31889",
"split": "train"
} |
math | [
{
"content": "Given a geometric sequence $\\{a_n\\}$ composed of positive numbers, with a common ratio $q=2$, and $a_1 \\cdot a_2 \\cdot a_3 \\cdot \\ldots \\cdot a_{30} = 2^{45}$, calculate the product of the terms $a_1 \\cdot a_4 \\cdot a_7 \\cdot \\ldots \\cdot a_{28}$. Please output the final answer within ... | {
"ground_truth": "2^5",
"style": "rule"
} | {
"index": "31890",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of all nonzero real numbers. Let $f : S \\to S$ be a function such that\n\\[f(x) + f(y) = f(xyf(x + y))\\]for all $x,$ $y \\in S$ such that $x + y \\neq 0.$\n\nLet $n$ be the number of possible values of $f(4),$ and let $s$ be the sum of all possible values of $f(4).$ Find $n... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "31891",
"split": "train"
} |
math | [
{
"content": "The arithmetic mean of the nine numbers in the set $\\{9, 99, 999, 9999, \\ldots, 999999999\\}$ is a $9$-digit number $M$, all of whose digits are distinct. The number $M$ doesn't contain the digit Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "31892",
"split": "train"
} |
math | [
{
"content": "Find the integer $n$, $0 \\le n \\le 6$, such that \\[n \\equiv 100000 \\pmod{7}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "31893",
"split": "train"
} |
math | [
{
"content": "Acute-angled $\\triangle ABC$ is inscribed in a circle with center at $O$. The measures of arcs are $\\stackrel \\frown {AB} = 80^\\circ$ and $\\stackrel \\frown {BC} = 100^\\circ$. A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. Find the ratio of the magnitudes of ... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31894",
"split": "train"
} |
math | [
{
"content": "Let $N$ be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to $N$, independently and uniformly at random. Let $p_{N}$ denote the probability that the product of these two integers has a units digit of 0. The maximum possible value of $p_{N}$ over all... | {
"ground_truth": "2800",
"style": "rule"
} | {
"index": "31895",
"split": "train"
} |
math | [
{
"content": "If $x \\cdot \\log_{27} 64 = 1$, then $4^x + 4^{-x} =$ \\_\\_\\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{10}{3}",
"style": "rule"
} | {
"index": "31896",
"split": "train"
} |
math | [
{
"content": "Point P moves on the parabola $y^2=4x$, and point Q moves on the line $x-y+5=0$. Find the minimum value of the sum of the distance $d$ from point P to the directrix of the parabola and the distance $|PQ|$ between points P and Q. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "3\\sqrt{2}",
"style": "rule"
} | {
"index": "31897",
"split": "train"
} |
math | [
{
"content": "Let $\\alpha$ and $\\beta$ be complex numbers such that $|\\beta| = 1$ and $\\overline{\\alpha} \\beta \\neq 1.$ Find the maximum value of\n\\[\\left| \\frac{\\beta - \\alpha}{1 - \\overline{\\alpha} \\beta} \\right|.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "31898",
"split": "train"
} |
math | [
{
"content": "For a four-digit natural number $M$, if the digit in the thousands place is $6$ more than the digit in the units place, and the digit in the hundreds place is $2$ more than the digit in the tens place, then $M$ is called a \"naive number.\" For example, the four-digit number $7311$ is a \"naive nu... | {
"ground_truth": "9313",
"style": "rule"
} | {
"index": "31899",
"split": "train"
} |
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