data_source stringclasses 4
values | prompt listlengths 1 1 | ability stringclasses 1
value | rank float64 0 1 | reward_model dict |
|---|---|---|---|---|
math | [
{
"content": "$x$ is a real number with the property that $x+\\tfrac1x = 3$. Let $S_m = x^m + \\tfrac{1}{x^m}$. Determine the value of $S_7$.\n",
"role": "user"
}
] | math | 0.187323 | {
"ground_truth": "843",
"style": "rule"
} |
amc | [
{
"content": "What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$?",
"role": "user"
}
] | math | 0.4435 | {
"ground_truth": "48",
"style": "rule"
} |
omni_math | [
{
"content": "Consider functions $f : [0, 1] \\rightarrow \\mathbb{R}$ which satisfy\n (i) for all in , (ii) , (iii) whenever , , and are all in .\nFind, with proof, the smallest constant $c$ such that\n$f(x) \\le cx$\nfor every function $f$ satisfying (i)-(iii) and every $x$ in $[0, 1]$ .",
"role": ... | math | 0.578084 | {
"ground_truth": "The smallest constant \\( c \\) such that \\( f(x) \\le cx \\) for every function \\( f \\) satisfying the given conditions is \\( c = 2 \\).",
"style": "rule"
} |
amc | [
{
"content": "In the country of East Westmore, statisticians estimate there is a baby born every $8$ hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year?",
"role": "user"
}
] | math | 0.057134 | {
"ground_truth": "700",
"style": "rule"
} |
omni_math | [
{
"content": "Let $A B C$ be a triangle where $A B=9, B C=10, C A=17$. Let $\\Omega$ be its circumcircle, and let $A_{1}, B_{1}, C_{1}$ be the diametrically opposite points from $A, B, C$, respectively, on $\\Omega$. Find the area of the convex hexagon with the vertices $A, B, C, A_{1}, B_{1}, C_{1}$.",
"ro... | math | 0.757887 | {
"ground_truth": "\\frac{1155}{4}",
"style": "rule"
} |
omni_math | [
{
"content": "How many six-digit multiples of 27 have only 3, 6, or 9 as their digits?",
"role": "user"
}
] | math | 0.917375 | {
"ground_truth": "51",
"style": "rule"
} |
amc | [
{
"content": "The expressions $a+bc$ and $(a+b)(a+c)$ are:",
"role": "user"
}
] | math | 0.640981 | {
"ground_truth": "equal whenever $a+b+c=1$",
"style": "rule"
} |
aime | [
{
"content": "Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$.",
"role": "user"
}
] | math | 0.855552 | {
"ground_truth": "42",
"style": "rule"
} |
math | [
{
"content": "An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the square of the length of each side is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",... | math | 0.195527 | {
"ground_truth": "937",
"style": "rule"
} |
omni_math | [
{
"content": "Find all positive real numbers $t$ with the following property: there exists an infinite set $X$ of real numbers such that the inequality \\[ \\max\\{|x-(a-d)|,|y-a|,|z-(a+d)|\\}>td\\] holds for all (not necessarily distinct) $x,y,z\\in X$, all real numbers $a$ and all positive real numbers $d$.",... | math | 0.774001 | {
"ground_truth": "t < \\frac{1}{2}",
"style": "rule"
} |
omni_math | [
{
"content": "Let $b(n)$ be the number of digits in the base -4 representation of $n$. Evaluate $\\sum_{i=1}^{2013} b(i)$.",
"role": "user"
}
] | math | 0.757105 | {
"ground_truth": "12345",
"style": "rule"
} |
aime | [
{
"content": "A strictly increasing sequence of positive integers $a_1$, $a_2$, $a_3$, $\\cdots$ has the property that for every positive integer $k$, the subsequence $a_{2k-1}$, $a_{2k}$, $a_{2k+1}$ is geometric and the subsequence $a_{2k}$, $a_{2k+1}$, $a_{2k+2}$ is arithmetic. Suppose that $a_{13} = 2016$. F... | math | 0.959859 | {
"ground_truth": "504",
"style": "rule"
} |
omni_math | [
{
"content": "In how many ways can we fill the cells of a $4\\times4$ grid such that each cell contains exactly one positive integer and the product of the numbers in each row and each column is $2020$?",
"role": "user"
}
] | math | 0.574372 | {
"ground_truth": "576",
"style": "rule"
} |
math | [
{
"content": "Bill can buy jags, jigs and jogs for $\\$1$, $\\$2$ and $\\$7$ each, respectively. What is the largest number of jogs he can purchase if he must buy at least one of each item and will spend exactly $\\$50$?",
"role": "user"
}
] | math | 0.384315 | {
"ground_truth": "6",
"style": "rule"
} |
math | [
{
"content": "For each integer $n$ greater than 1, let $F(n)$ be the number of solutions of the equation $\\sin x = \\sin nx$ on the interval $[0, \\pi]$. What is $\\sum_{n=2}^{2007} F(n)$?",
"role": "user"
}
] | math | 0.795976 | {
"ground_truth": "2{,}016{,}532",
"style": "rule"
} |
aime | [
{
"content": "Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$",
"role": "user"
}
] | math | 0.946186 | {
"ground_truth": "676",
"style": "rule"
} |
math | [
{
"content": "What is the smallest positive integer that is both a multiple of $7$ and a multiple of $4$?",
"role": "user"
}
] | math | 0.021193 | {
"ground_truth": "28",
"style": "rule"
} |
omni_math | [
{
"content": "Chris and Paul each rent a different room of a hotel from rooms $1-60$. However, the hotel manager mistakes them for one person and gives \"Chris Paul\" a room with Chris's and Paul's room concatenated. For example, if Chris had 15 and Paul had 9, \"Chris Paul\" has 159. If there are 360 rooms in ... | math | 0.805938 | {
"ground_truth": "\\frac{153}{1180}",
"style": "rule"
} |
amc | [
{
"content": "Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?\n\n[asy]\nunitsize(6mm);\ndefaultpen(linewidth(.8pt));\npath p=(1,1)--(-2,1)--(... | math | 0.389491 | {
"ground_truth": "3",
"style": "rule"
} |
math | [
{
"content": "Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. Find the number of such distinct triangles whose area is a positive integer.\n",
"role": "user"
}
] | math | 0.351792 | {
"ground_truth": "600",
"style": "rule"
} |
math | [
{
"content": "Let $n = 2^4 \\cdot 3^5 \\cdot 4^6\\cdot 6^7$. How many natural-number factors does $n$ have?",
"role": "user"
}
] | math | 0.145522 | {
"ground_truth": "312",
"style": "rule"
} |
math | [
{
"content": "Circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\\omega_1$, $\\omega_2$, and $\\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_i... | math | 0.50542 | {
"ground_truth": "552",
"style": "rule"
} |
math | [
{
"content": "A line through the points $(2, -9)$ and $(j, 17)$ is parallel to the line $2x + 3y = 21$. What is the value of $j$?",
"role": "user"
}
] | math | 0.155582 | {
"ground_truth": "-37",
"style": "rule"
} |
amc | [
{
"content": "Ann made a $3$-step staircase using $18$ toothpicks as shown in the figure. How many toothpicks does she need to add to complete a $5$-step staircase?\n[asy]\nsize(150);\ndefaultpen(linewidth(0.8));\npath h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45);\nfor(int i=0;i<=2;i=i+1) {\... | math | 0.498193 | {
"ground_truth": "22",
"style": "rule"
} |
omni_math | [
{
"content": "Find all functions $f : \\mathbb{Z}^+ \\to \\mathbb{Z}^+$ (where $\\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m - n$ divides $f(m) - f(n)$ for all distinct positive integers $m$ , $n$ .",
"role": "user"
}
] | math | 0.770095 | {
"ground_truth": "\\[\n\\boxed{f(n)=1, f(n)=2, f(n)=n}\n\\]",
"style": "rule"
} |
math | [
{
"content": "We can write\n\\[\\sum_{k = 1}^{100} (-1)^k \\cdot \\frac{k^2 + k + 1}{k!} = \\frac{a}{b!} - c,\\]where $a,$ $b,$ and $c$ are positive integers. Find the smallest possible value of $a + b + c.$",
"role": "user"
}
] | math | 0.600547 | {
"ground_truth": "202",
"style": "rule"
} |
aime | [
{
"content": "Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2 \\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3.$ One way to do this is shown below. Find the number of positive integer... | math | 0.744995 | {
"ground_truth": "144",
"style": "rule"
} |
omni_math | [
{
"content": "The altitudes of a triangle are $12$, $15$, and $20$. What is the area of this triangle?",
"role": "user"
}
] | math | 0.439984 | {
"ground_truth": "150",
"style": "rule"
} |
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).\\]",
"role": "user"
}
] | math | 0.518019 | {
"ground_truth": "-2036162",
"style": "rule"
} |
math | [
{
"content": "Two circles are centered at the origin, as shown. The point $P(8,6)$ is on the larger circle and the point $S(0,k)$ is on the smaller circle. If $QR=3$, what is the value of $k$?\n\n[asy]\nunitsize(0.2 cm);\n\ndefaultpen(linewidth(.7pt)+fontsize(10pt));\ndotfactor=4;\ndraw(Circle((0,0),7)); draw... | math | 0.271609 | {
"ground_truth": "7",
"style": "rule"
} |
amc | [
{
"content": "In $\\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\\overline{PQ}$, points $C$ and $D$ lie on $\\overline{QR}$, and points $E$ and $F$ lie on $\\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$.",
"role": "user"
}
] | math | 0.688544 | {
"ground_truth": "150",
"style": "rule"
} |
math | [
{
"content": "If $0 \\le p \\le 1$ and $0 \\le q \\le 1$, define $F(p, q)$ by\n\\[\n F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-q).\n\\]Define $G(p)$ to be the maximum of $F(p, q)$ over all $q$ (in the interval $0 \\le q \\le 1$). What is the value of $p$ (in the interval $0 \\le p \\le 1$) that minimizes ... | math | 0.197187 | {
"ground_truth": "\\frac{7}{12}",
"style": "rule"
} |
math | [
{
"content": "Rationalize the denominator of $\\frac{2}{3\\sqrt{5} + 2\\sqrt{11}}$ and write your answer in the form $\\displaystyle \\frac{A\\sqrt{B} + C\\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$?",
"role": "user"
}
] | math | 0.157633 | {
"ground_truth": "19",
"style": "rule"
} |
omni_math | [
{
"content": "Vova has a square grid $72\\times 72$. Unfortunately, $n$ cells are stained with coffee. Determine if Vova always can cut out a clean square $3\\times 3$ without its central cell, if \na) $n=699$; \nb) $n=750$.",
"role": "user"
}
] | math | 0.785331 | {
"ground_truth": "\\text{a) Yes, b) No}",
"style": "rule"
} |
amc | [
{
"content": "There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. What is it?",
"role": "user"
}
] | math | 0.503369 | {
"ground_truth": "7425",
"style": "rule"
} |
math | [
{
"content": "Find $x$ if\n\\[3 \\arctan \\frac{1}{4} + \\arctan \\frac{1}{20} + \\arctan \\frac{1}{x} = \\frac{\\pi}{4}.\\]",
"role": "user"
}
] | math | 0.320148 | {
"ground_truth": "1985",
"style": "rule"
} |
math | [
{
"content": "Convert $199_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in base 2. What is the value of $y-x?$",
"role": "user"
}
] | math | 0.079988 | {
"ground_truth": "2",
"style": "rule"
} |
amc | [
{
"content": "Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $32d$ in base $n$ equals $263$, and the value of the numeral $324$ in base $n$ equals the value of the numeral $11d1$ in base six. What is $n + d$?",
"role": "user"
}
] | math | 0.332747 | {
"ground_truth": "11",
"style": "rule"
} |
omni_math | [
{
"content": "Assume we are going to sell a bundle item including one unit of Item 1 and one unit of Item 2 at price p_{12}. The total cost of this item is t(c_{1} + c_{2}), where 0 < t < 1. Assume a visiting customer will purchase one piece of this bundle if (S_{1} + S_{2}) >= p_{12}, and if so, your profit is... | math | 0.915714 | {
"ground_truth": "p_{12}^{*} = \\begin{cases} \\frac{1}{3}(c_{12} + \\sqrt{c_{12}^{2} + 6u_{1}u_{2}}), & c_{12} \\in [0, \\frac{3}{2}u_{1} - u_{2}] \\\\ \\frac{1}{4}(u_{1} + 2u_{2} + 2c_{12}), & c_{12} \\in [\\frac{3}{2}u_{1} - u_{2}, u_{2} - \\frac{1}{2}u_{1}] \\\\ \\frac{1}{3}(u_{1} + u_{2} + 2c_{12}), & c_{12} \\... |
amc | [
{
"content": "For how many integers $x$ is the point $(x, -x)$ inside or on the circle of radius $10$ centered at $(5, 5)$?",
"role": "user"
}
] | math | 0.236254 | {
"ground_truth": "11",
"style": "rule"
} |
aime | [
{
"content": "Let $A_1,A_2,A_3,\\cdots,A_{12}$ be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set $\\{A_1,A_2,A_3,\\cdots,A_{12}\\} ?$",
"role": "user"
}
] | math | 0.785819 | {
"ground_truth": "183",
"style": "rule"
} |
omni_math | [
{
"content": "How many positive integers $n$ with $n \\leq 100$ can be expressed as the sum of four or more consecutive positive integers?",
"role": "user"
}
] | math | 0.733568 | {
"ground_truth": "63",
"style": "rule"
} |
omni_math | [
{
"content": "Consider a square, inside which is inscribed a circle, inside which is inscribed a square, inside which is inscribed a circle, and so on, with the outermost square having side length 1. Find the difference between the sum of the areas of the squares and the sum of the areas of the circles.",
"... | math | 0.355113 | {
"ground_truth": "2 - \\frac{\\pi}{2}",
"style": "rule"
} |
math | [
{
"content": "Let $N$ be the number of ways to write $2010$ in the form $2010 = a_3 \\cdot 10^3 + a_2 \\cdot 10^2 + a_1 \\cdot 10 + a_0$, where the $a_i$'s are integers, and $0 \\le a_i \\le 99$. An example of such a representation is $1\\cdot 10^3 + 3\\cdot 10^2 + 67\\cdot 10^1 + 40\\cdot 10^0$. Find $N$.\n",
... | math | 0.880555 | {
"ground_truth": "202",
"style": "rule"
} |
aime | [
{
"content": "Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form\n\\[12x^2 + bxy + cy^2 + d = 0.\\]\n Find the product $bc$.",
"role": "user"
}
] | math | 0.528274 | {
"ground_truth": "84",
"style": "rule"
} |
amc | [
{
"content": "The rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\\overline{AC}$ and $\\overline{AE}$, respectively. The area of quadrilateral $ABDF$ is",
"role": "user"
}
] | math | 0.380701 | {
"ground_truth": "320",
"style": "rule"
} |
omni_math | [
{
"content": "Find all functions $f : \\mathbb{R} \\to\\mathbb{R}$ such that $f(0)\\neq 0$ and\n\\[f(f(x)) + f(f(y)) = f(x + y)f(xy),\\]\nfor all $x, y \\in\\mathbb{R}$.",
"role": "user"
}
] | math | 0.398476 | {
"ground_truth": "{f(x)=2 \\; \\forall x \\in \\mathbb R}",
"style": "rule"
} |
math | [
{
"content": "In triangle $ABC$, $3 \\sin A + 4 \\cos B = 6$ and $4 \\sin B + 3 \\cos A = 1$. Find all possible values of $\\angle C,$ in degrees. Enter all the possible values, separated by commas.",
"role": "user"
}
] | math | 0.6109 | {
"ground_truth": "30^\\circ",
"style": "rule"
} |
math | [
{
"content": "The points $A$, $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$. It is given that $AB=13$, $BC=14$, $CA=15$, and that the distance from $O$ to $\\triangle ABC$ is $\\frac{m\\sqrt{n}}k$, where $m$, $n$, and $k$ are positive integers, $m$ and $k$ are relatively prime, and ... | math | 0.518508 | {
"ground_truth": "118",
"style": "rule"
} |
aime | [
{
"content": "Let $S$ be the set of positive integers $k$ such that the two parabolas\\[y=x^2-k~~\\text{and}~~x=2(y-20)^2-k\\]intersect in four distinct points, and these four points lie on a circle with radius at most $21$. Find the sum of the least element of $S$ and the greatest element of $S$.\nDiagram\n Gr... | math | 0.950679 | {
"ground_truth": "285",
"style": "rule"
} |
omni_math | [
{
"content": "Find all functions $ f: \\mathbb{Q}^{\\plus{}} \\mapsto \\mathbb{Q}^{\\plus{}}$ such that: \n\\[ f(x) \\plus{} f(y) \\plus{} 2xy f(xy) \\equal{} \\frac {f(xy)}{f(x\\plus{}y)}.\\]",
"role": "user"
}
] | math | 0.313996 | {
"ground_truth": "\\frac{1}{x^2}",
"style": "rule"
} |
aime | [
{
"content": "Let $(a_1,a_2,a_3,\\ldots,a_{12})$ be a permutation of $(1,2,3,\\ldots,12)$ for which\n$a_1>a_2>a_3>a_4>a_5>a_6 \\mathrm{\\ and \\ } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}.$\n An example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12).$ Find the number of such permutations.",
"role": "use... | math | 0.725754 | {
"ground_truth": "462",
"style": "rule"
} |
math | [
{
"content": "Completely factor the following expression: $9x^2+3x$.",
"role": "user"
}
] | math | 0.031253 | {
"ground_truth": "3x(3x+1)",
"style": "rule"
} |
math | [
{
"content": "If $x - y = 6$ and $x + y = 12$, what is the value of $x$?",
"role": "user"
}
] | math | 0.018947 | {
"ground_truth": "9",
"style": "rule"
} |
omni_math | [
{
"content": "Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane $90^\\circ$ counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$ to the triang... | math | 0.273367 | {
"ground_truth": "$\\text { No }$",
"style": "rule"
} |
aime | [
{
"content": "Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?",
"role": "user"
}
] | math | 0.376892 | {
"ground_truth": "120",
"style": "rule"
} |
amc | [
{
"content": "The locus of the centers of all circles of given radius $a$, in the same plane, passing through a fixed point, is:",
"role": "user"
}
] | math | 0.336752 | {
"ground_truth": "a circle",
"style": "rule"
} |
math | [
{
"content": "Let $x$ and $y$ be real numbers such that $2(x^2 + y^2) = x + y.$ Find the maximum value of $x - y.$",
"role": "user"
}
] | math | 0.318683 | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} |
aime | [
{
"content": "The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with th... | math | 0.214376 | {
"ground_truth": "732",
"style": "rule"
} |
aime | [
{
"content": "Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily i... | math | 0.611095 | {
"ground_truth": "114",
"style": "rule"
} |
amc | [
{
"content": "Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?",
"role": ... | math | 0.923528 | {
"ground_truth": "13",
"style": "rule"
} |
amc | [
{
"content": "In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?\n[... | math | 0.775076 | {
"ground_truth": "12",
"style": "rule"
} |
math | [
{
"content": "Compute\n\n$\\frac{1622^2-1615^2}{1629^2-1608^2}$.",
"role": "user"
}
] | math | 0.11466 | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} |
omni_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.",
"role": "user"
}
] | math | 0.569099 | {
"ground_truth": "(2n+1)(n+1)",
"style": "rule"
} |
math | [
{
"content": "How many solutions does the equation \\[\\frac{(x-1)(x-2)(x-3)\\dotsm(x-100)}{(x-1^2)(x-2^2)(x-3^2)\\dotsm(x-100^2)} = 0\\]have for $x$?",
"role": "user"
}
] | math | 0.230296 | {
"ground_truth": "90",
"style": "rule"
} |
omni_math | [
{
"content": "A sequence of positive integers $a_{1}, a_{2}, a_{3}, \\ldots$ satisfies $$a_{n+1}=n\\left\\lfloor\\frac{a_{n}}{n}\\right\\rfloor+1$$ for all positive integers $n$. If $a_{30}=30$, how many possible values can $a_{1}$ take? (For a real number $x$, $\\lfloor x\\rfloor$ denotes the largest integer t... | math | 0.952827 | {
"ground_truth": "274",
"style": "rule"
} |
math | [
{
"content": "Let $f(x)=|2\\{x\\}-1|$ where $\\{x\\}$ denotes the fractional part of $x$. The number $n$ is the smallest positive integer such that the equation \\[nf(xf(x))=x\\]has at least $2012$ real solutions. What is $n$?\n\nNote: the fractional part of $x$ is a real number $y=\\{x\\}$ such that $0\\le y<1... | math | 0.889638 | {
"ground_truth": "32",
"style": "rule"
} |
math | [
{
"content": "Four concentric circles are drawn with radii of 1, 3, 5 and 7. The inner circle is painted black, the ring around it is white, the next ring is black and the outer ring is white. What is the ratio of the black area to the white area? Express your answer as a common fraction.",
"role": "user"
... | math | 0.229905 | {
"ground_truth": "\\frac{17}{32}",
"style": "rule"
} |
omni_math | [
{
"content": "There are two values of $k$ for which the equation $x^{2}+2kx+7k-10=0$ has two equal real roots (that is, has exactly one solution for $x$). What is the sum of these values of $k$?",
"role": "user"
}
] | math | 0.155484 | {
"ground_truth": "7",
"style": "rule"
} |
math | [
{
"content": "The sum of two numbers is $45$. Their difference is $3$. What is the lesser of the two numbers?",
"role": "user"
}
] | math | 0.027932 | {
"ground_truth": "21",
"style": "rule"
} |
omni_math | [
{
"content": "Let $A B C$ be a triangle with $A B=5, B C=8$, and $C A=7$. Let $\\Gamma$ be a circle internally tangent to the circumcircle of $A B C$ at $A$ which is also tangent to segment $B C. \\Gamma$ intersects $A B$ and $A C$ at points $D$ and $E$, respectively. Determine the length of segment $D E$.",
... | math | 0.8201 | {
"ground_truth": "$\\frac{40}{9}$",
"style": "rule"
} |
math | [
{
"content": "The fenced area of a yard is a 15-foot by 12-foot rectangular region with a 3-foot by 3-foot square cut out, as shown. What is the area of the region within the fence, in square feet?\n\n[asy]draw((0,0)--(16,0)--(16,12)--(28,12)--(28,0)--(60,0)--(60,48)--(0,48)--cycle);\nlabel(\"15'\",(30,48),N);\... | math | 0.072273 | {
"ground_truth": "171",
"style": "rule"
} |
math | [
{
"content": "Find $h(x)$, with terms in order of decreasing degree, if \\[9x^3-3x+1+h(x)=3x^2-5x+3.\\]",
"role": "user"
}
] | math | 0.056841 | {
"ground_truth": "-9x^3+3x^2-2x+2",
"style": "rule"
} |
omni_math | [
{
"content": "For $0 \\leq p \\leq 1/2$, let $X_1, X_2, \\dots$ be independent random variables such that \\[ X_i = \\begin{cases} 1 & \\mbox{with probability $p$,} \\\\ -1 & \\mbox{with probability $p$,} \\\\ 0 & \\mbox{with probability $1-2p$,} \\end{cases} \\] for all $i \\geq 1$. Given a positive integer $n... | math | 0.815216 | {
"ground_truth": "p \\leq 1/4",
"style": "rule"
} |
amc | [
{
"content": "Let $S$ be the set $\\{1,2,3,...,19\\}$. For $a,b \\in S$, define $a \\succ b$ to mean that either $0 < a - b \\le 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \\succ y$, $y \\succ z$, and $z \\succ x$?",
"role": "user"
}
] | math | 0.707882 | {
"ground_truth": "855",
"style": "rule"
} |
amc | [
{
"content": "What is the largest quotient that can be formed using two numbers chosen from the set $\\{ -24, -3, -2, 1, 2, 8 \\}$?",
"role": "user"
}
] | math | 0.29749 | {
"ground_truth": "12",
"style": "rule"
} |
math | [
{
"content": "Find the remainder when $$88134 + 88135 + 88136 + 88137 + 88138 + 88139$$is divided by 9.",
"role": "user"
}
] | math | 0.242113 | {
"ground_truth": "6",
"style": "rule"
} |
math | [
{
"content": "Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of t... | math | 0.466256 | {
"ground_truth": "135",
"style": "rule"
} |
omni_math | [
{
"content": "In a photograph, Aristotle, David, Flora, Munirah, and Pedro are seated in a random order in a row of 5 chairs. If David is seated in the middle of the row, what is the probability that Pedro is seated beside him?",
"role": "user"
}
] | math | 0.207442 | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} |
math | [
{
"content": "Let $A := \\mathbb{Q} \\setminus \\{0,1\\}$ denote the set of all rationals other than 0 and 1. A function $f : A \\rightarrow \\mathbb{R}$ has the property that for all $x \\in A$,\n\\[\nf\\left( x\\right) + f\\left( 1 - \\frac{1}{x}\\right) = \\log\\lvert x\\rvert.\n\\]Compute the value of $f(20... | math | 0.621057 | {
"ground_truth": "\\log\\left(\\frac{2007}{2006}\\right)",
"style": "rule"
} |
amc | [
{
"content": "Consider the set of all fractions $\\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\\%$?",
"role": "user"
}
] | math | 0.40502 | {
"ground_truth": "1",
"style": "rule"
} |
math | [
{
"content": "Let $A=(0,9)$ and $B=(0,12)$. Points $A'$ and $B'$ are on the line $y=x$, and $\\overline{AA'}$ and $\\overline{BB'}$ intersect at $C=(2,8)$. What is the length of $\\overline{A'B'}$?",
"role": "user"
}
] | math | 0.278738 | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} |
aime | [
{
"content": "In a circle of radius $42$, two chords of length $78$ intersect at a point whose distance from the center is $18$. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed u... | math | 0.958297 | {
"ground_truth": "378",
"style": "rule"
} |
amc | [
{
"content": "Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana?",
"role": "user"
}
] | math | 0.242309 | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} |
math | [
{
"content": "In Mr. Abraham's class, $10$ of the $15$ students received an $A$ on the latest exam. If the same ratio of students received an $A$ on Mrs. Berkeley's latest exam, and if Mrs. Berkeley has $24$ students total, how many students in Mrs. Berkeley's class received an $A$?",
"role": "user"
}
] | math | 0.021779 | {
"ground_truth": "16",
"style": "rule"
} |
math | [
{
"content": "What is the sum of all the integers between -12.1 and 3.3?",
"role": "user"
}
] | math | 0.064655 | {
"ground_truth": "-72",
"style": "rule"
} |
omni_math | [
{
"content": "If $3^{x}=5$, what is the value of $3^{x+2}$?",
"role": "user"
}
] | math | 0.014748 | {
"ground_truth": "45",
"style": "rule"
} |
math | [
{
"content": "The expression $16x^2-106x-105$ can be written as $(8x + a)(2x + b),$ where $a$ and $b$ are integers. What is $a + 2b$?",
"role": "user"
}
] | math | 0.272976 | {
"ground_truth": "-23",
"style": "rule"
} |
omni_math | [
{
"content": "If $\\frac{1}{9}+\\frac{1}{18}=\\frac{1}{\\square}$, what is the number that replaces the $\\square$ to make the equation true?",
"role": "user"
}
] | math | 0.070417 | {
"ground_truth": "6",
"style": "rule"
} |
omni_math | [
{
"content": "Let $r_{1}, \\ldots, r_{n}$ be the distinct real zeroes of the equation $x^{8}-14 x^{4}-8 x^{3}-x^{2}+1=0$. Evaluate $r_{1}^{2}+\\cdots+r_{n}^{2}$",
"role": "user"
}
] | math | 0.60963 | {
"ground_truth": "8",
"style": "rule"
} |
omni_math | [
{
"content": "Find all polynomials $P$ with real coefficients such that \\[\\frac{P(x)}{yz}+\\frac{P(y)}{zx}+\\frac{P(z)}{xy}=P(x-y)+P(y-z)+P(z-x)\\] holds for all nonzero real numbers $x,y,z$ satisfying $2xyz=x+y+z$ .",
"role": "user"
}
] | math | 0.83514 | {
"ground_truth": "\\[ P(x) = c(x^2 + 3) \\text{ for any constant } c. \\]",
"style": "rule"
} |
amc | [
{
"content": "When Cheenu was a boy, he could run $15$ miles in $3$ hours and $30$ minutes. As an old man, he can now walk $10$ miles in $4$ hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy?",
"role": "user"
}
] | math | 0.235375 | {
"ground_truth": "10",
"style": "rule"
} |
aime | [
{
"content": "A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the ... | math | 0.718625 | {
"ground_truth": "810",
"style": "rule"
} |
math | [
{
"content": "How many real numbers $x$ are solutions to the following equation? \\[ |x-1| = |x-2| + |x-3| \\]",
"role": "user"
}
] | math | 0.391054 | {
"ground_truth": "2",
"style": "rule"
} |
amc | [
{
"content": "The six-digit number $20210A$ is prime for only one digit $A.$ What is $A?$",
"role": "user"
}
] | math | 0.462838 | {
"ground_truth": "9",
"style": "rule"
} |
omni_math | [
{
"content": "Let $S=\\{-100,-99,-98, \\ldots, 99,100\\}$. Choose a 50-element subset $T$ of $S$ at random. Find the expected number of elements of the set $\\{|x|: x \\in T\\}$.",
"role": "user"
}
] | math | 0.522707 | {
"ground_truth": "\\frac{8825}{201}",
"style": "rule"
} |
amc | [
{
"content": "A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral?",
"role": "user"
}
] | math | 0.745288 | {
"ground_truth": "18",
"style": "rule"
} |
aime | [
{
"content": "Call a positive integer $n$ $k$-pretty if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$-pretty. Let $S$ be the sum of positive integers less than $2019$ that are $20$-pretty. Find $\\tfrac{S}{20}$.",
"role": "user"
}
] | math | 0.530521 | {
"ground_truth": "472",
"style": "rule"
} |
omni_math | [
{
"content": "Calculate the value of $\\frac{2 \\times 3 + 4}{2 + 3}$.",
"role": "user"
}
] | math | 0.00879 | {
"ground_truth": "2",
"style": "rule"
} |
omni_math | [
{
"content": "For a positive integer $n$, let $p(n)$ denote the product of the positive integer factors of $n$. Determine the number of factors $n$ of 2310 for which $p(n)$ is a perfect square.",
"role": "user"
}
] | math | 0.701533 | {
"ground_truth": "27",
"style": "rule"
} |
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