data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given that Fox wants to ensure he has 20 coins left after crossing the bridge four times, and paying a $50$-coin toll each time, determine the number of coins that Fox had at the beginning. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "4100",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[\\begin{vmatrix} 7 & 3 \\\\ -1 & 2 \\end{vmatrix}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "4101",
"split": "train"
} |
math | [
{
"content": "Let $\\#$ be the relation defined by $A \\# B = A^2 + B^2$. If $A \\# 5 = 169$, what is the positive value of $A$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "4102",
"split": "train"
} |
math | [
{
"content": "On each side of a right-angled triangle, a semicircle is drawn with that side as a diameter. The areas of the three semicircles are \\( x^{2} \\), \\( 3x \\), and 180, where \\( x^{2} \\) and \\( 3x \\) are both less than 180. What is the area of the smallest semicircle? Please output the final an... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "4103",
"split": "train"
} |
math | [
{
"content": "Let $b_n$ be the integer obtained by writing down the integers from 1 to $n$ in reverse, from right to left. Compute the remainder when $b_{39}$ is divided by 125. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "4104",
"split": "train"
} |
math | [
{
"content": "Two boys and three girls stand in a row for a photo. If boy A does not stand at either end, and exactly two of the three girls are adjacent, determine the number of different arrangements. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "4105",
"split": "train"
} |
math | [
{
"content": "If the polynomial $x^3+x^{10}=a_0+a_1(x+1)+\\ldots+a_9(x+1)^9+a_{10}(x+1)^{10}$, then $a_2=$ ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "4106",
"split": "train"
} |
math | [
{
"content": "Determine the smallest positive integer \\(n\\) for which there exists positive real numbers \\(a\\) and \\(b\\) such that\n\\[(a + 3bi)^n = (a - 3bi)^n,\\]\nand compute \\(\\frac{b}{a}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "4107",
"split": "train"
} |
math | [
{
"content": "Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\\neq0$ such that\n(i) $m<2a$;\n(ii) $2n|(2am-m^2+n^2)$;\n(iii) $n^2-m^2+2mn\\leq2a(n-m)$.\nFor $(m, n)\\in A$, let \\[f(m,n)=\\frac{2am-m^2-mn}{n}.\\]\nDetermine the maximum and minimum values of $f$. Please output the fin... | {
"ground_truth": "2 \\text{ and } 3750",
"style": "rule"
} | {
"index": "4108",
"split": "train"
} |
math | [
{
"content": "How many distinct three-digit positive integers have only odd digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "4109",
"split": "train"
} |
math | [
{
"content": "The blue parabola shown is the graph of the equation \\( x = ay^2 + by + c \\). The vertex of the parabola is at \\( (5, 3) \\), and it passes through the point \\( (3, 5) \\). Find \\( c \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "4110",
"split": "train"
} |
math | [
{
"content": "Given a line $y = \\frac{\\sqrt{3}}{3}x$ and a circle $C$ with its center on the positive x-axis and a radius of 2 intersects the line at points $A$ and $B$ such that $|AB|=2\\sqrt{3}$.\n\n(1) Given a point $P(-1, \\sqrt{7})$, and $Q$ is any point on circle $C$, find the maximum value of $|PQ|$.\n... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "4111",
"split": "train"
} |
math | [
{
"content": "Two more than three times $B$ is equal to 20. What is the value of $B$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "4112",
"split": "train"
} |
math | [
{
"content": "What is the 43rd digit after the decimal point in the decimal representation of $\\frac{1}{13}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "4113",
"split": "train"
} |
math | [
{
"content": "Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\\{ 1,2,\\dots, 1000\\}$ there exist two sets, whose intersection contains at least $m$ numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "4114",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle in which (${BL}$is the angle bisector of ${\\angle{ABC}}$ $\\left( L\\in AC \\right)$, ${AH}$ is an altitude of$\\vartriangle ABC$ $\\left( H\\in BC \\right)$ and ${M}$is the midpoint of the side ${AB}$. It is known that the midpoints of the segments ${BL}$ and ${MH}$ coin... | {
"ground_truth": "60^\\circ",
"style": "rule"
} | {
"index": "4115",
"split": "train"
} |
math | [
{
"content": "An ice cream shop offers 8 different flavors of ice cream. What is the greatest number of sundaes that can be made if each sundae can consist of 1, 2, or 3 scoops, with each scoop possibly being a different type of ice cream and no two sundaes having the same combination of flavors? Please output ... | {
"ground_truth": "92",
"style": "rule"
} | {
"index": "4116",
"split": "train"
} |
math | [
{
"content": "Two sectors of a circle of radius $12$ are placed side by side, as shown. Determine the $\\textit{area}$ of figure $ABCD.$ [asy]\ndraw((0,0)--(12,0)..(10.3923,6)..(6,10.3923)--(-6,10.3923)..(-4.3923,4.3923)..(0,0),black+linewidth(1));\ndraw((0,0)--(6,10.3923),black+linewidth(1)+dashed);\nlabel(\"$... | {
"ground_truth": "48\\pi",
"style": "rule"
} | {
"index": "4117",
"split": "train"
} |
math | [
{
"content": "Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\\triangle ABC$ is a right triangle with area $2008$. What is the sum of the digits of the $y$-coordinate of $C$? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "4118",
"split": "train"
} |
math | [
{
"content": "How many pairs of parallel edges, such as $\\overline{AB}$ and $\\overline{GH}$ or $\\overline{EH}$ and $\\overline{FG}$, does a cube have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "4119",
"split": "train"
} |
math | [
{
"content": "Find the number of functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[f(xy) + f(xz) - f(x) f(yz) \\ge 1\\]for all real numbers $x,$ $y,$ and $z.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "4120",
"split": "train"
} |
math | [
{
"content": "In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. Given that 2(tanA + tanB) = $\\frac{\\text{tanA}}{\\text{cosB}} + \\frac{\\text{tanB}}{\\text{cosA}}$.\n(1) Find the value of $\\frac{a+b}{c}$;\n(2) If c = 2 and C = $\\frac{\\pi}{3}$, find the area of triangle... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "4121",
"split": "train"
} |
math | [
{
"content": "Let \\\\(\\triangle ABC\\\\) have internal angles \\\\(A\\\\), \\\\(B\\\\), and \\\\(C\\\\) opposite to sides of lengths \\\\(a\\\\), \\\\(b\\\\), and \\\\(c\\\\) respectively, and it satisfies \\\\(a^{2}+c^{2}-b^{2}= \\sqrt {3}ac\\\\). \n\\\\((1)\\\\) Find the size of angle \\\\(B\\\\); \n\\\\(... | {
"ground_truth": "\\sqrt {3}",
"style": "rule"
} | {
"index": "4122",
"split": "train"
} |
math | [
{
"content": "Let $a_0$, $a_1$, $a_2$, $\\dots$ be an infinite sequence of real numbers such that $a_0 = \\frac{5}{13}$ and\n\\[\n a_{n} = 2 a_{n-1}^2 - 1\n\\]for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies t... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "4123",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a\\_n\\}$, for any $k \\in \\mathbb{N}^*$, when $n = 3k$, $a\\_n = a\\_{\\frac{n}{3}}$; when $n \\neq 3k$, $a\\_n = n$. The 10th occurrence of 2 in this sequence is the \\_\\_\\_\\_\\_\\_th term. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\cdot 3^{9}",
"style": "rule"
} | {
"index": "4124",
"split": "train"
} |
math | [
{
"content": "In a regular decagon $ABCDEFGHIJ$, points $K$, $L$, $M$, $N$, $O$, $P$, $Q$, $R$, and $S$ are selected on the sides $\\overline{AB}$, $\\overline{BC}$, $\\overline{CD}$, $\\overline{DE}$, $\\overline{EF}$, $\\overline{FG}$, $\\overline{GH}$, $\\overline{HI}$, and $\\overline{IJ}$ respectively. Eac... | {
"ground_truth": "\\frac{3\\sqrt{3}}{40}",
"style": "rule"
} | {
"index": "4125",
"split": "train"
} |
math | [
{
"content": "$r(x)$ has domain $\\{-1,0,1,2\\}$ and range $\\{0,2,4,6\\}$. $s(x)$ has domain $\\{1,2,3,4\\}$ and is defined by $s(x)=x+1$. What is the sum of all possible values of $s(r(x))$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "4126",
"split": "train"
} |
math | [
{
"content": "Let $a_1, a_2, \\ldots$ and $b_1, b_2, \\ldots$ be arithmetic progressions such that $a_1 = 50, b_1 = 100$, and $a_{50} + b_{50} = 850$. Find the sum of the first fifty terms of the progression $a_1 + b_1, a_2 + b_2, \\ldots$ Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "25000",
"style": "rule"
} | {
"index": "4127",
"split": "train"
} |
math | [
{
"content": "Bill draws two circles which intersect at $X,Y$ . Let $P$ be the intersection of the common tangents to the two circles and let $Q$ be a point on the line segment connecting the centers of the two circles such that lines $PX$ and $QX$ are perpendicular. Given that the radii of the two cir... | {
"ground_truth": "4807",
"style": "rule"
} | {
"index": "4128",
"split": "train"
} |
math | [
{
"content": "If the line $y=kx+t$ is a tangent line to the curve $y=e^x+2$ and also a tangent line to the curve $y=e^{x+1}$, find the value of $t$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4-2\\ln 2",
"style": "rule"
} | {
"index": "4129",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle with $AB=5, BC=4$ and $AC=3$. Let $\\mathcal{P}$ and $\\mathcal{Q}$ be squares inside $ABC$ with disjoint interiors such that they both have one side lying on $AB$. Also, the two squares each have an edge lying on a common line perpendicular to $AB$, and $\\mathcal{P}$ has ... | {
"ground_truth": "\\frac{144}{49}",
"style": "rule"
} | {
"index": "4130",
"split": "train"
} |
math | [
{
"content": "Write a twelve-digit number that is not a perfect cube. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100000000000",
"style": "rule"
} | {
"index": "4131",
"split": "train"
} |
math | [
{
"content": "Given $tan({θ+\\frac{π}{4}})=2tanθ-7$, determine the value of $\\sin 2\\theta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "4132",
"split": "train"
} |
math | [
{
"content": "Given functions $f(x)=-2x$ for $x<0$ and $g(x)=\\frac{x}{\\ln x}+x-2$. If $f(x_{1})=g(x_{2})$, find the minimum value of $x_{2}-2x_{1}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{e}-2",
"style": "rule"
} | {
"index": "4133",
"split": "train"
} |
math | [
{
"content": "In a two-day problem-solving competition, Gamma and Delta participated and attempted questions worth a total of 500 points. On the first day, Gamma scored 180 points out of 280 points attempted, and on the second day, he scored 120 points out of 220 points attempted. Delta, who also divided his at... | {
"ground_truth": "\\frac{409}{500}",
"style": "rule"
} | {
"index": "4134",
"split": "train"
} |
math | [
{
"content": "Suppose the function $f(x) = ax + \\frac{x}{x-1}$ where $x > 1$.\n(1) If $a > 0$, find the minimum value of the function $f(x)$.\n(2) If $a$ is chosen from the set \\{1, 2, 3\\} and $b$ is chosen from the set \\{2, 3, 4, 5\\}, find the probability that $f(x) > b$ always holds true. Please output t... | {
"ground_truth": "\\frac{5}{6}",
"style": "rule"
} | {
"index": "4135",
"split": "train"
} |
math | [
{
"content": "How many ways can you arrange 15 dominoes (after removing all dominoes with five or six pips) in a single line according to the usual rules of the game, considering arrangements from left to right and right to left as different? As always, the dominoes must be placed such that matching pips (e.g.,... | {
"ground_truth": "126760",
"style": "rule"
} | {
"index": "4136",
"split": "train"
} |
math | [
{
"content": "In triangle $XYZ$, $\\angle X = 90^\\circ$ and $\\sin Y = \\frac{3}{5}$. Find $\\cos Z$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "4137",
"split": "train"
} |
math | [
{
"content": "The perimeter of a semicircle with an area of ______ square meters is 15.42 meters. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14.13",
"style": "rule"
} | {
"index": "4138",
"split": "train"
} |
math | [
{
"content": "Rectangle \\( EFGH \\) is 10 cm by 6 cm. \\( P \\) is the midpoint of \\( \\overline{EF} \\), and \\( Q \\) is the midpoint of \\( \\overline{GH} \\). Calculate the area of region \\( EPGQ \\).\n\n** Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "4139",
"split": "train"
} |
math | [
{
"content": "Compute $\\tan \\left (\\operatorname{arccot} \\frac{4}{7} \\right).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{4}",
"style": "rule"
} | {
"index": "4140",
"split": "train"
} |
math | [
{
"content": "A certain company, in response to the national call for garbage classification, has launched a project with the support of the research department to innovate technologies. The project processes kitchen waste into reusable chemical products. It is known that the daily processing capacity $x$ (unit... | {
"ground_truth": "1800",
"style": "rule"
} | {
"index": "4141",
"split": "train"
} |
math | [
{
"content": "$2018$ people (call them $A, B, C, \\ldots$ ) stand in a line with each permutation equally likely. Given that $A$ stands before $B$ , what is the probability that $C$ stands after $B$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/3",
"style": "rule"
} | {
"index": "4142",
"split": "train"
} |
math | [
{
"content": "Let \\( p, q, r, s \\) be distinct real numbers such that the roots of \\( x^2 - 12px - 13q = 0 \\) are \\( r \\) and \\( s \\), and the roots of \\( x^2 - 12rx - 13s = 0 \\) are \\( p \\) and \\( q \\). Find the value of \\( p + q + r + s \\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "2028",
"style": "rule"
} | {
"index": "4143",
"split": "train"
} |
math | [
{
"content": "What is the diameter in centimeters of a circle whose area is $100\\pi \\text{cm}^2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "4144",
"split": "train"
} |
math | [
{
"content": "Three real numbers $x, y, z$ are chosen randomly, and independently of each other, between 0 and 1, inclusive. What is the probability that each of $x-y$ and $x-z$ is greater than $-\\frac{1}{2}$ and less than $\\frac{1}{2}$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{7}{12}",
"style": "rule"
} | {
"index": "4145",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{P}$ be the parabola given by the equation \\( y = x^2 \\). Suppose a circle $\\mathcal{C}$ intersects $\\mathcal{P}$ at four distinct points. If three of these points are \\((-4,16)\\), \\((1,1)\\), and \\((6,36)\\), find the sum of the distances from the directrix of the parabola t... | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "4146",
"split": "train"
} |
math | [
{
"content": "Real numbers \\(a, b, c\\) and positive number \\(\\lambda\\) make the function \\(f(x) = x^3 + ax^2 + bx + c\\) have three real roots \\(x_1, x_2, x_3\\), such that (1) \\(x_2 - x_1 = \\lambda\\); (2) \\(x_3 > \\frac{1}{2}(x_1 + x_2)\\). Find the maximum value of \\(\\frac{2a^3 + 27c + 9ab}{\\lam... | {
"ground_truth": "\\frac{3\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "4147",
"split": "train"
} |
math | [
{
"content": "Beatriz loves odd numbers. How many numbers between 0 and 1000 can she write using only odd digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "155",
"style": "rule"
} | {
"index": "4148",
"split": "train"
} |
math | [
{
"content": "A and B play a game with the following rules: In the odd-numbered rounds, A has a winning probability of $\\frac{3}{4}$, and in the even-numbered rounds, B has a winning probability of $\\frac{3}{4}$. There are no ties in any round, and the game ends when one person has won 2 more rounds than the ... | {
"ground_truth": "16/3",
"style": "rule"
} | {
"index": "4149",
"split": "train"
} |
math | [
{
"content": "The price of an article was increased $p\\%$. Later the new price was decreased $p\\%$. If the last price was one dollar, the original price was: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10000}{10000-p^2}",
"style": "rule"
} | {
"index": "4150",
"split": "train"
} |
math | [
{
"content": "At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to b... | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "4151",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has vertices $A(-2, 10)$, $B(3, 0)$, $C(10, 0)$. A line through $B$ cuts the area of $\\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-10",
"style": "rule"
} | {
"index": "4152",
"split": "train"
} |
math | [
{
"content": "Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005.$ Find $\\lfloor P\\rfloor.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "4153",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{D}$ be the $2 \\times 2$ matrix corresponding to the dilation, centered at the origin, with scale factor 7. Find $\\det \\mathbf{D}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "4154",
"split": "train"
} |
math | [
{
"content": "Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling. Fox is excited about his good fortune until he discovers that all his mo... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "4155",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate plane, a polar coordinate system is established with the origin as the pole and the non-negative half of the x-axis as the polar axis. It is known that point A has polar coordinates $$( \\sqrt{2}, \\frac{\\pi}{4})$$, and the parametric equation of line $l$ is:\n$$\\begi... | {
"ground_truth": "\\frac{4\\sqrt{2}}{5}",
"style": "rule"
} | {
"index": "4156",
"split": "train"
} |
math | [
{
"content": "Vendelín lives between two bus stops, at three-eighths of their distance. Today he left home and discovered that whether he ran to one or the other stop, he would arrive at the stop at the same time as the bus. The average speed of the bus is $60 \\mathrm{~km} / \\mathrm{h}$.\n\nWhat is the averag... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "4157",
"split": "train"
} |
math | [
{
"content": "A hollow silver sphere with an outer diameter of $2 R = 1 \\mathrm{dm}$ is exactly half-submerged in water. What is the thickness of the sphere's wall if the specific gravity of silver is $s = 10.5$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.008",
"style": "rule"
} | {
"index": "4158",
"split": "train"
} |
math | [
{
"content": "We color certain squares of an $8 \\times 8$ chessboard red. How many squares can we color at most if we want no red trimino? How many squares can we color at least if we want every trimino to have at least one red square? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "4159",
"split": "train"
} |
math | [
{
"content": "To celebrate 2019, Faraz gets four sandwiches shaped in the digits 2, 0, 1, and 9 at lunch. However, the four digits get reordered (but not flipped or rotated) on his plate and he notices that they form a 4-digit multiple of 7. What is the greatest possible number that could have been formed? Plea... | {
"ground_truth": "1092",
"style": "rule"
} | {
"index": "4160",
"split": "train"
} |
math | [
{
"content": "Distribute 7 students into two dormitories, A and B, with each dormitory having at least 2 students. How many different distribution plans are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "4161",
"split": "train"
} |
math | [
{
"content": "Given that $a_{1}, a_{2}, \\cdots, a_{10}$ are ten different positive integers satisfying the equation $\\left|a_{i+1}-a_{i}\\right|=2 \\text { or } 3$, where $i=1,2, \\cdots, 10$, with the condition $a_{11}=a_{1}$, determine the maximum value of $M-m$, where $M$ is the maximum number among $a_{1... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "4162",
"split": "train"
} |
math | [
{
"content": "A factory produces two types of products, A and B, with profits P and Q (in ten thousand yuan), respectively. The relationship between the profits and the invested capital m (in ten thousand yuan) follows the empirical formulas P = (1/3)m + 65 and Q = 76 + 4√m. Now, 150 ten thousand yuan of capita... | {
"ground_truth": "203",
"style": "rule"
} | {
"index": "4163",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of all 3-digit numbers with all digits in the set $\\{1,2,3,4,5,6,7\\}$ (so in particular, all three digits are nonzero). For how many elements $\\overline{a b c}$ of $S$ is it true that at least one of the (not necessarily distinct) 'digit cycles' $\\overline{a b c}, \\overline... | {
"ground_truth": "127",
"style": "rule"
} | {
"index": "4164",
"split": "train"
} |
math | [
{
"content": "Given that the function $F(x) = f(x) + x^2$ is an odd function, and $f(2) = 1$, find $f(-2) = ( \\ )$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-9",
"style": "rule"
} | {
"index": "4165",
"split": "train"
} |
math | [
{
"content": "Let $N$ be the smallest positive integer such that $N+2N+3N+\\ldots +9N$ is a number all of whose digits are equal. What is the sum of digits of $N$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "4166",
"split": "train"
} |
math | [
{
"content": "Let $a>0$ and $b>0,$ and define two operations:\n$$a \\nabla b = \\dfrac{a + b}{1 + ab}$$\n$$a \\Delta b = \\dfrac{a - b}{1 - ab}$$\nCalculate $3 \\nabla 4$ and $3 \\Delta 4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{11}",
"style": "rule"
} | {
"index": "4167",
"split": "train"
} |
math | [
{
"content": "Given that the magnitude of vector $\\overrightarrow {a}$ is 1, the magnitude of vector $\\overrightarrow {b}$ is 2, and the magnitude of $\\overrightarrow {a}+ \\overrightarrow {b}$ is $\\sqrt {7}$, find the angle between $\\overrightarrow {a}$ and $\\overrightarrow {b}$. Please output the final ... | {
"ground_truth": "\\frac {\\pi}{3}",
"style": "rule"
} | {
"index": "4168",
"split": "train"
} |
math | [
{
"content": "By definition, a polygon is regular if all its angles and sides are equal. Points \\( A, B, C, D \\) are consecutive vertices of a regular polygon (in that order). It is known that the angle \\( ABD = 135^\\circ \\). How many vertices does this polygon have? Please output the final answer within \... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "4169",
"split": "train"
} |
math | [
{
"content": "Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their spot to the spectator (there were no draws). It turned out that Egor participated in 13 games, and Nikita in 27 games. How many games did Innokentiy play? Please ... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "4170",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a trapezoid with $AB \\parallel CD$, $AB=11$, $BC=5$, $CD=19$, and $DA=7$. Bisectors of $\\angle A$ and $\\angle D$ meet at $P$, and bisectors of $\\angle B$ and $\\angle C$ meet at $Q$. What is the area of hexagon $ABQCDP$? Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "$30\\sqrt{3}$",
"style": "rule"
} | {
"index": "4171",
"split": "train"
} |
math | [
{
"content": "Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{149}{12}",
"style": "rule"
} | {
"index": "4172",
"split": "train"
} |
math | [
{
"content": "Compute \\( 105 \\times 95 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9975",
"style": "rule"
} | {
"index": "4173",
"split": "train"
} |
math | [
{
"content": "Two differentiable real functions \\( f(x) \\) and \\( g(x) \\) satisfy\n\\[ \\frac{f^{\\prime}(x)}{g^{\\prime}(x)} = e^{f(x) - g(x)} \\]\nfor all \\( x \\), and \\( f(0) = g(2003) = 1 \\). Find the largest constant \\( c \\) such that \\( f(2003) > c \\) for all such functions \\( f, g \\). Pleas... | {
"ground_truth": "1 - \\ln 2",
"style": "rule"
} | {
"index": "4174",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=2\\ln x - ax^2 + 3$, \n(1) Discuss the monotonicity of the function $y=f(x)$; \n(2) If there exist real numbers $m, n \\in [1, 5]$ such that $f(m)=f(n)$ holds when $n-m \\geq 2$, find the maximum value of the real number $a$. Please output the final answer within \\boxed... | {
"ground_truth": "\\frac{\\ln 3}{4}",
"style": "rule"
} | {
"index": "4175",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy: $|\\overrightarrow{a}| = |\\overrightarrow{b}| = 1$, and $|k\\overrightarrow{a} + \\overrightarrow{b}| = \\sqrt{3}|\\overrightarrow{a} - k\\overrightarrow{b}| (k > 0)$. Find the maximum value of the angle between vectors $... | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "4176",
"split": "train"
} |
math | [
{
"content": "For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "4177",
"split": "train"
} |
math | [
{
"content": "The point $A$ $(3,4)$ is reflected over the $x$-axis to $B$. Then $B$ is reflected over the line $y=x$ to $C$. What is the area of triangle $ABC$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "4178",
"split": "train"
} |
math | [
{
"content": "Given the parabola $C: x^{2}=2py\\left(p \\gt 0\\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\\left(y+4\\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ ... | {
"ground_truth": "20\\sqrt{5}",
"style": "rule"
} | {
"index": "4179",
"split": "train"
} |
math | [
{
"content": "We plotted the graph of the function \\( f(x) = \\frac{1}{x} \\) in the coordinate system. How should we choose the new, still equal units on the axes, if we want the curve to become the graph of the function \\( g(x) = \\frac{2}{x} \\)? Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "4180",
"split": "train"
} |
math | [
{
"content": "How many numbers are in the list $250, 243, 236, \\ldots, 29, 22?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "4181",
"split": "train"
} |
math | [
{
"content": "In a $10 \\times 5$ grid, an ant starts from point $A$ and can only move right or up along the grid lines but is not allowed to pass through point $C$. How many different paths are there from point $A$ to point $B$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1827",
"style": "rule"
} | {
"index": "4182",
"split": "train"
} |
math | [
{
"content": "Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Fi... | {
"ground_truth": "929",
"style": "rule"
} | {
"index": "4183",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive value of $x$ such that $x + 4321$ results in a palindrome? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "4184",
"split": "train"
} |
math | [
{
"content": "On a blackboard, the number 123456789 is written. Select two adjacent digits from this number, and if neither of them is 0, subtract 1 from each and swap their positions. For example: \\( 123456789 \\rightarrow 123436789 \\rightarrow \\cdots \\). After performing this operation several times, what... | {
"ground_truth": "101010101",
"style": "rule"
} | {
"index": "4185",
"split": "train"
} |
math | [
{
"content": "Calculate: $(128)^{\\frac{7}{3}}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "65536 \\cdot \\sqrt[3]{2}",
"style": "rule"
} | {
"index": "4186",
"split": "train"
} |
math | [
{
"content": "In how many ways can the set of ordered pairs of integers be colored red and blue such that for all $a$ and $b$, the points $(a, b),(-1-b, a+1)$, and $(1-b, a-1)$ are all the same color? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "4187",
"split": "train"
} |
math | [
{
"content": "The equations\n\\[60x^4 + ax^3 + bx^2 + cx + 20 = 0\\]and\n\\[20x^5 + dx^4 + ex^3 + fx^2 + gx + 60 = 0\\]have a common rational root $r$ which is not an integer, and which is positive. What is $r?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "4188",
"split": "train"
} |
math | [
{
"content": "Circles $\\omega_1$ and $\\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\\omega$ is externally tangent to both $\\omega_1$ and $\\omega_2$. Suppose line $AB$ intersects $\\omega$ at two points $P$ and $Q$ such that the measure of mino... | {
"ground_truth": "672",
"style": "rule"
} | {
"index": "4189",
"split": "train"
} |
math | [
{
"content": "Peter has three times as many sisters as brothers. His sister Louise has twice as many sisters as brothers. How many children are there in the family? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "4190",
"split": "train"
} |
math | [
{
"content": "Subtract $123.45$ from $567.89.$ Express the result as a decimal to the nearest hundredth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "444.44",
"style": "rule"
} | {
"index": "4191",
"split": "train"
} |
math | [
{
"content": "If $x \\geq 0$, then $\\sqrt{x\\sqrt{x\\sqrt{x}}} =$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\sqrt[8]{x^7}$",
"style": "rule"
} | {
"index": "4192",
"split": "train"
} |
math | [
{
"content": "The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum? Pleas... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "4193",
"split": "train"
} |
math | [
{
"content": "The spinner shown is divided into 6 sections of equal size. Determine the probability of landing on a section that contains the letter Q using this spinner. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{6}",
"style": "rule"
} | {
"index": "4194",
"split": "train"
} |
math | [
{
"content": "A right triangle when rotating around a large leg forms a cone with a volume of $100\\pi$ . Calculate the length of the path that passes through each vertex of the triangle at rotation of $180^o$ around the point of intersection of its bisectors, if the sum of the diameters of the circles, insc... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "4195",
"split": "train"
} |
math | [
{
"content": "How many distinct solutions are there to the equation $|x-7| = |x+1|$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "4196",
"split": "train"
} |
math | [
{
"content": "The local junior football team is deciding on their new uniforms. The team's ninth-graders will choose the color of the socks (options: red, green, or blue), and the tenth-graders will pick the color for the t-shirts (options: red, yellow, green, blue, or white). Neither group will discuss their c... | {
"ground_truth": "\\frac{13}{15}",
"style": "rule"
} | {
"index": "4197",
"split": "train"
} |
math | [
{
"content": "Given the right focus $F$ and the right directrix $l$ of the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, with eccentricity $e = \\frac{\\sqrt{5}}{5}$. Draw $AM \\perp l$ through the vertex $A(0, b)$, with $M$ as the foot of the perpendicular. Find the slope of the line $FM$. Ple... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "4198",
"split": "train"
} |
math | [
{
"content": "A cauldron has the shape of a paraboloid of revolution. The radius of its base is \\( R = 3 \\) meters, and the depth is \\( H = 5 \\) meters. The cauldron is filled with a liquid, the specific weight of which is \\( 0.8 \\Gamma / \\text{cm}^3 \\). Calculate the work required to pump the liquid ou... | {
"ground_truth": "294300\\pi",
"style": "rule"
} | {
"index": "4199",
"split": "train"
} |
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