context_messages listlengths 1 1 | ground_truth stringlengths 1 3 | meta dict |
|---|---|---|
[
{
"content": "Question:\nFor polynomial $P(x)=1-\\dfrac{1}{3}x+\\dfrac{1}{6}x^{2}$ , define $Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\\sum_{i=0}^{50} a_ix^{i}$ .\nThen $\\sum_{i=0}^{50} |a_i|=\\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .\nPlease reason step by step, a... | 275 | {
"ability": "math",
"data_source": "aime",
"index": 700,
"split": "train"
} |
[
{
"content": "Question:\nSquares $ABCD$ and $EFGH$ have a common center and $\\overline{AB} || \\overline{EF}$ . The area of $ABCD$ is 2016, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on ... | 840 | {
"ability": "math",
"data_source": "aime",
"index": 701,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of sets $\\{a,b,c\\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,$ and $61$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 728 | {
"ability": "math",
"data_source": "aime",
"index": 702,
"split": "train"
} |
[
{
"content": "Question:\nThe sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$ . There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$ . Find $c_k$ .\nPlease reason step by... | 262 | {
"ability": "math",
"data_source": "aime",
"index": 703,
"split": "train"
} |
[
{
"content": "Question:\nTriangle $ABC$ is inscribed in circle $\\omega$ . Points $P$ and $Q$ are on side $\\overline{AB}$ with $AP<AQ$ . Rays $CP$ and $CQ$ meet $\\omega$ again at $S$ and $T$ (other than $C$ ), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$ , then $ST=\\frac{m}{n}$ , where $m$ and $n$ are ... | 43 | {
"ability": "math",
"data_source": "aime",
"index": 704,
"split": "train"
} |
[
{
"content": "Question:\nFor positive integers $N$ and $k$ , define $N$ to be $k$ -nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$ -nice nor $8$ -nice.\nPlease reason step by step, and put... | 749 | {
"ability": "math",
"data_source": "aime",
"index": 705,
"split": "train"
} |
[
{
"content": "Question:\nThe 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 pain... | 732 | {
"ability": "math",
"data_source": "aime",
"index": 706,
"split": "train"
} |
[
{
"content": "Question:\nBeatrix is going to place six rooks on a $6 \\times 6$ chessboard where both the rows and columns are labeled $1$ to $6$ ; the rooks are placed so that no two rooks are in the same row or the same column. The value of a square is the sum of its row number and column number. The score of... | 371 | {
"ability": "math",
"data_source": "aime",
"index": 707,
"split": "train"
} |
[
{
"content": "Question:\nEquilateral $\\triangle ABC$ 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 ang... | 450 | {
"ability": "math",
"data_source": "aime",
"index": 708,
"split": "train"
} |
[
{
"content": "Question:\nFor $1 \\leq i \\leq 215$ let $a_i = \\dfrac{1}{2^{i}}$ and $a_{216} = \\dfrac{1}{2^{215}}$ . Let $x_1, x_2, ..., x_{216}$ be positive real numbers such that $\\sum_{i=1}^{216} x_i=1$ and $\\sum_{1 \\leq i < j \\leq 216} x_ix_j = \\dfrac{107}{215} + \\sum_{i=1}^{216} \\dfrac{a_i x_i^{2}... | 863 | {
"ability": "math",
"data_source": "aime",
"index": 709,
"split": "train"
} |
[
{
"content": "Question:\nFifteen distinct points are designated on $\\triangle ABC$ : the 3 vertices $A$ , $B$ , and $C$ ; $3$ other points on side $\\overline{AB}$ ; $4$ other points on side $\\overline{BC}$ ; and $5$ other points on side $\\overline{CA}$ . Find the number of triangles with positive area whose... | 390 | {
"ability": "math",
"data_source": "aime",
"index": 710,
"split": "train"
} |
[
{
"content": "Question:\nWhen each of $702$ , $787$ , and $855$ is divided by the positive integer $m$ , the remainder is always the positive integer $r$ . When each of $412$ , $722$ , and $815$ is divided by the positive integer $n$ , the remainder is always the positive integer $s \\neq r$ . Find $m+n+r+s$ .\... | 62 | {
"ability": "math",
"data_source": "aime",
"index": 711,
"split": "train"
} |
[
{
"content": "Question:\nFor a positive integer $n$ , let $d_n$ be the units digit of $1 + 2 + \\dots + n$ . Find the remainder when \\[\\sum_{n=1}^{2017} d_n\\] is divided by $1000$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 69 | {
"ability": "math",
"data_source": "aime",
"index": 712,
"split": "train"
} |
[
{
"content": "Question:\nA pyramid has a triangular base with side lengths $20$ , $20$ , and $24$ . The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$ . The volume of the pyramid is $m\\sqrt{n}$ , where $m$ and $n$ are positive integers, an... | 803 | {
"ability": "math",
"data_source": "aime",
"index": 713,
"split": "train"
} |
[
{
"content": "Question:\nA rational number written in base eight is $\\underline{a} \\underline{b} . \\underline{c} \\underline{d}$ , where all digits are nonzero. The same number in base twelve is $\\underline{b} \\underline{b} . \\underline{b} \\underline{a}$ . Find the base-ten number $\\underline{a} \\under... | 321 | {
"ability": "math",
"data_source": "aime",
"index": 714,
"split": "train"
} |
[
{
"content": "Question:\nA circle circumscribes an isosceles triangle whose two congruent angles have degree measure $x$ . Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\\frac{14}{25}$ . Fi... | 48 | {
"ability": "math",
"data_source": "aime",
"index": 715,
"split": "train"
} |
[
{
"content": "Question:\nFor nonnegative integers $a$ and $b$ with $a + b \\leq 6$ , let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$ . Let $S$ denote the sum of all $T(a, b)$ , where $a$ and $b$ are nonnegative integers with $a + b \\leq 6$ . Find the remainder when $S$ is divided by $1000$ .\nPle... | 564 | {
"ability": "math",
"data_source": "aime",
"index": 716,
"split": "train"
} |
[
{
"content": "Question:\nTwo real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$ . Let $O$ and $P$ be two points on the plane with $OP = 200$ . Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\\angle POQ$ and $\\angle POR$ are ... | 41 | {
"ability": "math",
"data_source": "aime",
"index": 717,
"split": "train"
} |
[
{
"content": "Question:\nLet $a_{10} = 10$ , and for each positive integer $n >10$ let $a_n = 100a_{n - 1} + n$ . Find the least positive $n > 10$ such that $a_n$ is a multiple of $99$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 45 | {
"ability": "math",
"data_source": "aime",
"index": 718,
"split": "train"
} |
[
{
"content": "Question:\nLet $z_1 = 18 + 83i$ , $z_2 = 18 + 39i,$ and $z_3 = 78 + 99i,$ where $i = \\sqrt{-1}$ . Let $z$ be the unique complex number with the properties that $\\frac{z_3 - z_1}{z_2 - z_1} \\cdot \\frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. F... | 56 | {
"ability": "math",
"data_source": "aime",
"index": 719,
"split": "train"
} |
[
{
"content": "Question:\nConsider arrangements of the $9$ numbers $1, 2, 3, \\dots, 9$ in a $3 \\times 3$ array. For each such arrangement, let $a_1$ , $a_2$ , and $a_3$ be the medians of the numbers in rows $1$ , $2$ , and $3$ respectively, and let $m$ be the median of $\\{a_1, a_2, a_3\\}$ . Let $Q$ be the nu... | 360 | {
"ability": "math",
"data_source": "aime",
"index": 720,
"split": "train"
} |
[
{
"content": "Question:\nCall a set $S$ product-free if there do not exist $a, b, c \\in S$ (not necessarily distinct) such that $a b = c$ . For example, the empty set and the set $\\{16, 20\\}$ are product-free, whereas the sets $\\{4, 16\\}$ and $\\{2, 8, 16\\}$ are not product-free. Find the number of produc... | 252 | {
"ability": "math",
"data_source": "aime",
"index": 721,
"split": "train"
} |
[
{
"content": "Question:\nFor every $m \\geq 2$ , let $Q(m)$ be the least positive integer with the following property: For every $n \\geq Q(m)$ , there is always a perfect cube $k^3$ in the range $n < k^3 \\leq mn$ . Find the remainder when \\[\\sum_{m = 2}^{2017} Q(m)\\] is divided by $1000$ .\nPlease reason s... | 59 | {
"ability": "math",
"data_source": "aime",
"index": 722,
"split": "train"
} |
[
{
"content": "Question:\nLet $a > 1$ and $x > 1$ satisfy $\\log_a(\\log_a(\\log_a 2) + \\log_a 24 - 128) = 128$ and $\\log_a(\\log_a x) = 256$ . Find the remainder when $x$ is divided by $1000$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 896 | {
"ability": "math",
"data_source": "aime",
"index": 723,
"split": "train"
} |
[
{
"content": "Question:\nThe area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths $2\\sqrt3$ , $5$ , and $\\sqrt{37}$ , as shown, is $\\tfrac{m\\sqrt{p}}{n}$ , where $m$ , $n$ , and $p$ are positive integers, $m$ and $n$ are relatively prime, and... | 145 | {
"ability": "math",
"data_source": "aime",
"index": 724,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of subsets of $\\{1, 2, 3, 4, 5, 6, 7, 8\\}$ that are subsets of neither $\\{1, 2, 3, 4, 5\\}$ nor $\\{4, 5, 6, 7, 8\\}$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 196 | {
"ability": "math",
"data_source": "aime",
"index": 725,
"split": "train"
} |
[
{
"content": "Question:\nTeams $T_1$ , $T_2$ , $T_3$ , and $T_4$ are in the playoffs. In the semifinal matches, $T_1$ plays $T_4$ , and $T_2$ plays $T_3$ . The winners of those two matches will play each other in the final match to determine the champion. When $T_i$ plays $T_j$ , the probability that $T_i$ wins... | 781 | {
"ability": "math",
"data_source": "aime",
"index": 726,
"split": "train"
} |
[
{
"content": "Question:\nA triangle has vertices $A(0,0)$ , $B(12,0)$ , and $C(8,10)$ . The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Fi... | 409 | {
"ability": "math",
"data_source": "aime",
"index": 727,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 222 | {
"ability": "math",
"data_source": "aime",
"index": 728,
"split": "train"
} |
[
{
"content": "Question:\nA set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189$ , $320$ , $287$ , $234$ , $x$ , and $y$ . Find the greatest possible value of $x+y$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"ro... | 791 | {
"ability": "math",
"data_source": "aime",
"index": 729,
"split": "train"
} |
[
{
"content": "Question:\nFind the sum of all positive integers $n$ such that $\\sqrt{n^2+85n+2017}$ is an integer.\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 195 | {
"ability": "math",
"data_source": "aime",
"index": 730,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\\log(kx)=2\\log(x+2)$ has exactly one real solution.\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 501 | {
"ability": "math",
"data_source": "aime",
"index": 731,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of positive integers $n$ less than $2017$ such that \\[1+n+\\frac{n^2}{2!}+\\frac{n^3}{3!}+\\frac{n^4}{4!}+\\frac{n^5}{5!}+\\frac{n^6}{6!}\\] is an integer.\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 134 | {
"ability": "math",
"data_source": "aime",
"index": 732,
"split": "train"
} |
[
{
"content": "Question:\nA special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card... | 13 | {
"ability": "math",
"data_source": "aime",
"index": 733,
"split": "train"
} |
[
{
"content": "Question:\nRectangle $ABCD$ has side lengths $AB=84$ and $AD=42$ . Point $M$ is the midpoint of $\\overline{AD}$ , point $N$ is the trisection point of $\\overline{AB}$ closer to $A$ , and point $O$ is the intersection of $\\overline{CM}$ and $\\overline{DN}$ . Point $P$ lies on the quadrilateral ... | 546 | {
"ability": "math",
"data_source": "aime",
"index": 734,
"split": "train"
} |
[
{
"content": "Question:\nFive towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing throug... | 544 | {
"ability": "math",
"data_source": "aime",
"index": 735,
"split": "train"
} |
[
{
"content": "Question:\nCircle $C_0$ has radius $1$ , and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$ . Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\\circ}$ counterclockwise from $A_0$ on $C_1$ . Circle $C_2$ has r... | 110 | {
"ability": "math",
"data_source": "aime",
"index": 736,
"split": "train"
} |
[
{
"content": "Question:\nFor each integer $n\\geq3$ , let $f(n)$ be the number of $3$ -element subsets of the vertices of a regular $n$ -gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$ .\nPlease reason step by ste... | 245 | {
"ability": "math",
"data_source": "aime",
"index": 737,
"split": "train"
} |
[
{
"content": "Question:\nA $10\\times10\\times10$ grid of points consists of all points in space of the form $(i,j,k)$ , where $i$ , $j$ , and $k$ are integers between $1$ and $10$ , inclusive. Find the number of different lines that contain exactly $8$ of these points.\nPlease reason step by step, and put your... | 168 | {
"ability": "math",
"data_source": "aime",
"index": 738,
"split": "train"
} |
[
{
"content": "Question:\nTetrahedron $ABCD$ has $AD=BC=28$ , $AC=BD=44$ , and $AB=CD=52$ . For any point $X$ in space, define $f(X)=AX+BX+CX+DX$ . The least possible value of $f(X)$ can be expressed as $m\\sqrt{n}$ , where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. F... | 682 | {
"ability": "math",
"data_source": "aime",
"index": 739,
"split": "train"
} |
[
{
"content": "Question:\nLet $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \\leq a \\leq 100$ and $b \\geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided b... | 600 | {
"ability": "math",
"data_source": "aime",
"index": 740,
"split": "train"
} |
[
{
"content": "Question:\nThe number $n$ can be written in base $14$ as $\\underline{a}\\text{ }\\underline{b}\\text{ }\\underline{c}$ , can be written in base $15$ as $\\underline{a}\\text{ }\\underline{c}\\text{ }\\underline{b}$ , and can be written in base $6$ as $\\underline{a}\\text{ }\\underline{c}\\text{ ... | 925 | {
"ability": "math",
"data_source": "aime",
"index": 741,
"split": "train"
} |
[
{
"content": "Question:\nKathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGG... | 157 | {
"ability": "math",
"data_source": "aime",
"index": 742,
"split": "train"
} |
[
{
"content": "Question:\nIn $\\triangle ABC, AB = AC = 10$ and $BC = 12$ . Point $D$ lies strictly between $A$ and $B$ on $\\overline{AB}$ and point $E$ lies strictly between $A$ and $C$ on $\\overline{AC}$ so that $AD = DE = EC$ . Then $AD$ can be expressed in the form $\\dfrac{p}{q}$ , where $p$ and $q$ are r... | 289 | {
"ability": "math",
"data_source": "aime",
"index": 743,
"split": "train"
} |
[
{
"content": "Question:\nFor each ordered pair of real numbers $(x,y)$ satisfying \\[\\log_2(2x+y) = \\log_4(x^2+xy+7y^2)\\] there is a real number $K$ such that \\[\\log_3(3x+y) = \\log_9(3x^2+4xy+Ky^2).\\] Find the product of all possible values of $K$ .\nPlease reason step by step, and put your final answer ... | 189 | {
"ability": "math",
"data_source": "aime",
"index": 744,
"split": "train"
} |
[
{
"content": "Question:\nLet $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 440 | {
"ability": "math",
"data_source": "aime",
"index": 745,
"split": "train"
} |
[
{
"content": "Question:\nA right hexagonal prism has height $2$ . The bases are regular hexagons with side length $1$ . Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).\nPlease reason step by step, and put your final answ... | 52 | {
"ability": "math",
"data_source": "aime",
"index": 746,
"split": "train"
} |
[
{
"content": "Question:\nLet $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$ , and $DE=12$ . Denote $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 147 | {
"ability": "math",
"data_source": "aime",
"index": 747,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of four-element subsets of $\\{1,2,3,4,\\dots, 20\\}$ with the property that two distinct elements of a subset have a sum of $16$ , and two distinct elements of a subset have a sum of $24$ . For example, $\\{3,5,13,19\\}$ and $\\{6,10,20,18\\}$ are two such subsets.\nPle... | 210 | {
"ability": "math",
"data_source": "aime",
"index": 748,
"split": "train"
} |
[
{
"content": "Question:\nThe wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point $A$ . At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner c... | 4 | {
"ability": "math",
"data_source": "aime",
"index": 749,
"split": "train"
} |
[
{
"content": "Question:\nFind the least positive integer $n$ such that when $3^n$ is written in base $143$ , its two right-most digits in base $143$ are $01$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 195 | {
"ability": "math",
"data_source": "aime",
"index": 750,
"split": "train"
} |
[
{
"content": "Question:\nFor every subset $T$ of $U = \\{ 1,2,3,\\ldots,18 \\}$ , let $s(T)$ be the sum of the elements of $T$ , with $s(\\emptyset)$ defined to be $0$ . If $T$ is chosen at random among all subsets of $U$ , the probability that $s(T)$ is divisible by $3$ is $\\frac{m}{n}$ , where $m$ and $n$ ar... | 683 | {
"ability": "math",
"data_source": "aime",
"index": 751,
"split": "train"
} |
[
{
"content": "Question:\nLet $\\triangle ABC$ have side lengths $AB=30$ , $BC=32$ , and $AC=34$ . Point $X$ lies in the interior of $\\overline{BC}$ , and points $I_1$ and $I_2$ are the incenters of $\\triangle ABX$ and $\\triangle ACX$ , respectively. Find the minimum possible area of $\\triangle AI_1I_2$ as $... | 126 | {
"ability": "math",
"data_source": "aime",
"index": 752,
"split": "train"
} |
[
{
"content": "Question:\nLet $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$ . From any vertex of the heptagon except $E$ , the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$ , the frog stops and stays there. Find the number of distinct sequences of jumps... | 351 | {
"ability": "math",
"data_source": "aime",
"index": 753,
"split": "train"
} |
[
{
"content": "Question:\nDavid found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A,\\text{ }B,\\text{ }C$ , which can each be inscribed in a circle with radius $1$ . Let $\\varphi_A$ denote the measure of the acute angle made by the diagonals of q... | 59 | {
"ability": "math",
"data_source": "aime",
"index": 754,
"split": "train"
} |
[
{
"content": "Question:\nPoints $A$ , $B$ , and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$ , P... | 800 | {
"ability": "math",
"data_source": "aime",
"index": 755,
"split": "train"
} |
[
{
"content": "Question:\nLet $a_{0} = 2$ , $a_{1} = 5$ , and $a_{2} = 8$ , and for $n > 2$ define $a_{n}$ recursively to be the remainder when $4(a_{n-1} + a_{n-2} + a_{n-3})$ is divided by $11$ . Find $a_{2018} \\cdot a_{2020} \\cdot a_{2022}$ .\nPlease reason step by step, and put your final answer within \\b... | 112 | {
"ability": "math",
"data_source": "aime",
"index": 756,
"split": "train"
} |
[
{
"content": "Question:\nFind the sum of all positive integers $b < 1000$ such that the base- $b$ integer $36_{b}$ is a perfect square and the base- $b$ integer $27_{b}$ is a perfect cube.\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 371 | {
"ability": "math",
"data_source": "aime",
"index": 757,
"split": "train"
} |
[
{
"content": "Question:\nIn equiangular octagon $CAROLINE$ , $CA = RO = LI = NE =$ $\\sqrt{2}$ and $AR = OL = IN = EC = 1$ . The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$ , that is, the total area of the six triangular region... | 23 | {
"ability": "math",
"data_source": "aime",
"index": 758,
"split": "train"
} |
[
{
"content": "Question:\nSuppose that $x$ , $y$ , and $z$ are complex numbers such that $xy = -80 - 320i$ , $yz = 60$ , and $zx = -96 + 24i$ , where $i$ $=$ $\\sqrt{-1}$ . Then there are real numbers $a$ and $b$ such that $x + y + z = a + bi$ . Find $a^2 + b^2$ .\nPlease reason step by step, and put your final ... | 74 | {
"ability": "math",
"data_source": "aime",
"index": 759,
"split": "train"
} |
[
{
"content": "Question:\nA real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$ . The probability that the roots of the polynomial \\[x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2\\] are all real can be written in the form $\\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive ... | 37 | {
"ability": "math",
"data_source": "aime",
"index": 760,
"split": "train"
} |
[
{
"content": "Question:\nTriangle $ABC$ has side lengths $AB = 9$ , $BC =$ $5\\sqrt{3}$ , and $AC = 12$ . Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2, ..., 2449$ , and points $A = Q_{0}, Q_{1}, Q_{2}, ... , Q_{24... | 20 | {
"ability": "math",
"data_source": "aime",
"index": 761,
"split": "train"
} |
[
{
"content": "Question:\nA frog is positioned at the origin of the coordinate plane. From the point $(x, y)$ , the frog can jump to any of the points $(x + 1, y)$ , $(x + 2, y)$ , $(x, y + 1)$ , or $(x, y + 2)$ . Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4... | 556 | {
"ability": "math",
"data_source": "aime",
"index": 762,
"split": "train"
} |
[
{
"content": "Question:\nOctagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing 6-8-10 triangles from the corners of a $23$ $\\times$ $27$ rectangle with side $\\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\\over... | 184 | {
"ability": "math",
"data_source": "aime",
"index": 763,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of functions $f(x)$ from $\\{1, 2, 3, 4, 5\\}$ to $\\{1, 2, 3, 4, 5\\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\\{1, 2, 3, 4, 5\\}$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 756 | {
"ability": "math",
"data_source": "aime",
"index": 764,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of permutations of $1, 2, 3, 4, 5, 6$ such that for each $k$ with $1$ $\\leq$ $k$ $\\leq$ $5$ , at least one of the first $k$ terms of the permutation is greater than $k$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}... | 461 | {
"ability": "math",
"data_source": "aime",
"index": 765,
"split": "train"
} |
[
{
"content": "Question:\nLet $ABCD$ be a convex quadrilateral with $AB = CD = 10$ , $BC = 14$ , and $AD = 2\\sqrt{65}$ . Assume that the diagonals of $ABCD$ intersect at point $P$ , and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$ . Find the are... | 112 | {
"ability": "math",
"data_source": "aime",
"index": 766,
"split": "train"
} |
[
{
"content": "Question:\nMisha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .\nPlease reason step by s... | 647 | {
"ability": "math",
"data_source": "aime",
"index": 767,
"split": "train"
} |
[
{
"content": "Question:\nThe incircle $\\omega$ of triangle $ABC$ is tangent to $\\overline{BC}$ at $X$ . Let $Y \\neq X$ be the other intersection of $\\overline{AX}$ with $\\omega$ . Points $P$ and $Q$ lie on $\\overline{AB}$ and $\\overline{AC}$ , respectively, so that $\\overline{PQ}$ is tangent to $\\omega... | 227 | {
"ability": "math",
"data_source": "aime",
"index": 768,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of functions $f$ from $\\{0, 1, 2, 3, 4, 5, 6\\}$ to the integers such that $f(0) = 0$ , $f(6) = 12$ , and \\[|x - y| \\leq |f(x) - f(y)| \\leq 3|x - y|\\] for all $x$ and $y$ in $\\{0, 1, 2, 3, 4, 5, 6\\}$ .\nPlease reason step by step, and put your final answer wit... | 185 | {
"ability": "math",
"data_source": "aime",
"index": 769,
"split": "train"
} |
[
{
"content": "Question:\nConsider the integer \\[N = 9 + 99 + 999 + 9999 + \\cdots + \\underbrace{99\\ldots 99}_\\text{321 digits}.\\] Find the sum of the digits of $N$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 342 | {
"ability": "math",
"data_source": "aime",
"index": 770,
"split": "train"
} |
[
{
"content": "Question:\nJenn randomly chooses a number $J$ from $1, 2, 3,\\ldots, 19, 20$ . Bela then randomly chooses a number $B$ from $1, 2, 3,\\ldots, 19, 20$ distinct from $J$ . The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\\tfrac{m}{n}$ , where $m$ and $n$ ar... | 29 | {
"ability": "math",
"data_source": "aime",
"index": 771,
"split": "train"
} |
[
{
"content": "Question:\nIn $\\triangle PQR$ , $PR=15$ , $QR=20$ , 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 $ABCDEF$ .\nPlease reason step by step... | 120 | {
"ability": "math",
"data_source": "aime",
"index": 772,
"split": "train"
} |
[
{
"content": "Question:\nA soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $3$ substitutions, where any one of the $11$ players in the game is replaced by one of the substitutes... | 122 | {
"ability": "math",
"data_source": "aime",
"index": 773,
"split": "train"
} |
[
{
"content": "Question:\nA moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$ , it moves at random to one of the points $(a-1,b)$ , $(a,b-1)$ , or $(a-1,b-1)$ , each with probability $\\tfrac{1}{3}$ , indep... | 252 | {
"ability": "math",
"data_source": "aime",
"index": 774,
"split": "train"
} |
[
{
"content": "Question:\nIn convex quadrilateral $KLMN$ , side $\\overline{MN}$ is perpendicular to diagonal $\\overline{KM}$ , side $\\overline{KL}$ is perpendicular to diagonal $\\overline{LN}$ , $MN = 65$ , and $KL = 28$ . The line through $L$ perpendicular to side $\\overline{KN}$ intersects diagonal $\\ove... | 90 | {
"ability": "math",
"data_source": "aime",
"index": 775,
"split": "train"
} |
[
{
"content": "Question:\nThere are positive integers $x$ and $y$ that satisfy the system of equations \\[\\log_{10} x + 2 \\log_{10} (\\gcd(x,y)) = 60\\] \\[\\log_{10} y + 2 \\log_{10} (\\text{lcm}(x,y)) = 570.\\] Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x... | 880 | {
"ability": "math",
"data_source": "aime",
"index": 776,
"split": "train"
} |
[
{
"content": "Question:\nLet $x$ be a real number such that $\\sin^{10}x+\\cos^{10} x = \\tfrac{11}{36}$ . Then $\\sin^{12}x+\\cos^{12} x = \\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role"... | 67 | {
"ability": "math",
"data_source": "aime",
"index": 777,
"split": "train"
} |
[
{
"content": "Question:\nLet $\\tau(n)$ denote the number of positive integer divisors of $n$ . Find the sum of the six least positive integers $n$ that are solutions to $\\tau (n) + \\tau (n+1) = 7$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 540 | {
"ability": "math",
"data_source": "aime",
"index": 778,
"split": "train"
} |
[
{
"content": "Question:\nFor distinct complex numbers $z_1,z_2,\\dots,z_{673}$ , the polynomial \\[(x-z_1)^3(x-z_2)^3 \\cdots (x-z_{673})^3\\] can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$ , where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$ . The value of \\[\... | 352 | {
"ability": "math",
"data_source": "aime",
"index": 779,
"split": "train"
} |
[
{
"content": "Question:\nIn $\\triangle ABC$ , the sides have integer lengths and $AB=AC$ . Circle $\\omega$ has its center at the incenter of $\\triangle ABC$ . An excircle of $\\triangle ABC$ is a circle in the exterior of $\\triangle ABC$ that is tangent to one side of the triangle and tangent to the extensi... | 20 | {
"ability": "math",
"data_source": "aime",
"index": 780,
"split": "train"
} |
[
{
"content": "Question:\nGiven $f(z) = z^2-19z$ , there are complex numbers $z$ with the property that $z$ , $f(z)$ , and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$ . There are positive integers $m$ and $n$ such that one such value of $z$ is $m+\\sqrt{n}+11i... | 230 | {
"ability": "math",
"data_source": "aime",
"index": 781,
"split": "train"
} |
[
{
"content": "Question:\nTriangle $ABC$ has side lengths $AB=4$ , $BC=5$ , and $CA=6$ . Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$ . The point $F \\neq C$ is a point of intersection of the circumcircles of $\\triangle ACD$ and $\\triangle EBC$ satisfying $DF=2$ and $EF=7$ . Then $BE$ can be expressed as... | 32 | {
"ability": "math",
"data_source": "aime",
"index": 782,
"split": "train"
} |
[
{
"content": "Question:\nFind the least odd prime factor of $2019^8 + 1$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 97 | {
"ability": "math",
"data_source": "aime",
"index": 783,
"split": "train"
} |
[
{
"content": "Question:\nLet $\\overline{AB}$ be a chord of a circle $\\omega$ , and let $P$ be a point on the chord $\\overline{AB}$ . Circle $\\omega_1$ passes through $A$ and $P$ and is internally tangent to $\\omega$ . Circle $\\omega_2$ passes through $B$ and $P$ and is internally tangent to $\\omega$ . Ci... | 65 | {
"ability": "math",
"data_source": "aime",
"index": 784,
"split": "train"
} |
[
{
"content": "Question:\nTwo different points, $C$ and $D$ , lie on the same side of line $AB$ so that $\\triangle ABC$ and $\\triangle BAD$ are congruent with $AB=9,BC=AD=10$ , and $CA=DB=17$ . The intersection of these two triangular regions has area $\\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime po... | 59 | {
"ability": "math",
"data_source": "aime",
"index": 785,
"split": "train"
} |
[
{
"content": "Question:\nLily pads $1,2,3,\\ldots$ lie in a row on a pond. A frog makes a sequence of jumps starting on pad $1$ . From any pad $k$ the frog jumps to either pad $k+1$ or pad $k+2$ chosen randomly with probability $\\tfrac{1}{2}$ and independently of other jumps. The probability that the frog visi... | 107 | {
"ability": "math",
"data_source": "aime",
"index": 786,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of $7$ -tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following system of equations: \\[abc=70\\] \\[cde=71\\] \\[efg=72.\\]\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 96 | {
"ability": "math",
"data_source": "aime",
"index": 787,
"split": "train"
} |
[
{
"content": "Question:\nA standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .\nPlease reason step by step, and put your final answer within \... | 187 | {
"ability": "math",
"data_source": "aime",
"index": 788,
"split": "train"
} |
[
{
"content": "Question:\nFour ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order $1$ to $12$ . Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the $8$ peo... | 520 | {
"ability": "math",
"data_source": "aime",
"index": 789,
"split": "train"
} |
[
{
"content": "Question:\nIn a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$ , for some fixed $b\\ge2$ . A Martian student writes down \\[3\\log(\\sqrt{x}\\log x)=56\\] \\[\\log_{\\log x}(x)=54\\] and finds that this system of equations has a single real number sol... | 216 | {
"ability": "math",
"data_source": "aime",
"index": 790,
"split": "train"
} |
[
{
"content": "Question:\nTriangle $ABC$ has side lengths $AB=120,BC=220$ , and $AC=180$ . Lines $\\ell_A,\\ell_B$ , and $\\ell_C$ are drawn parallel to $\\overline{BC},\\overline{AC}$ , and $\\overline{AB}$ , respectively, such that the intersections of $\\ell_A,\\ell_B$ , and $\\ell_C$ with the interior of $\\... | 715 | {
"ability": "math",
"data_source": "aime",
"index": 791,
"split": "train"
} |
[
{
"content": "Question:\nThe polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019,$ and $f\\left(\\tfrac{1+\\sqrt3i}{2}\\right)=2015+2019\\sqrt3i$ . Find the remainder when $f(1)$ is divided by $1000$ .\nPlease reason step by step, and put your final answer within \\boxed{ }... | 53 | {
"ability": "math",
"data_source": "aime",
"index": 792,
"split": "train"
} |
[
{
"content": "Question:\nCall 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 the positive integers less than $2019$ that are $20$ -pretty. Find $\\tfrac{S}{20}$ .\nPlease reason step by step, and ... | 472 | {
"ability": "math",
"data_source": "aime",
"index": 793,
"split": "train"
} |
[
{
"content": "Question:\nThere is a unique angle $\\theta$ between $0^\\circ$ and $90^\\circ$ such that for nonnegative integers $n,$ the value of $\\tan(2^n\\theta)$ is positive when $n$ is a multiple of $3$ , and negative otherwise. The degree measure of $\\theta$ is $\\tfrac{p}{q}$ , where $p$ and $q$ are re... | 547 | {
"ability": "math",
"data_source": "aime",
"index": 794,
"split": "train"
} |
[
{
"content": "Question:\nTriangle $ABC$ has side lengths $AB=7,BC=8,$ and $CA=9.$ Circle $\\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\\omega_1$ and $\\omega_2$ not equal to $A... | 11 | {
"ability": "math",
"data_source": "aime",
"index": 795,
"split": "train"
} |
[
{
"content": "Question:\nFor $n\\ge1$ call a finite sequence $(a_1,a_2,\\ldots,a_n)$ of positive integers progressive if $a_i<a_{i+1}$ and $a_i$ divides $a_{i+1}$ for $1\\le i\\le n-1$ . Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360.$\nPlease reason ste... | 47 | {
"ability": "math",
"data_source": "aime",
"index": 796,
"split": "train"
} |
[
{
"content": "Question:\nRegular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1.$ Point $P$ lies inside the circle so that the region bounded by $\\overline{PA_1},\\overline{PA_2},$ and the minor arc $\\widehat{A_1A_2}$ of the circle has area $\\tfrac{1}{7},$ while the region bounded by ... | 504 | {
"ability": "math",
"data_source": "aime",
"index": 797,
"split": "train"
} |
[
{
"content": "Question:\nFind the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed.\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
... | 71 | {
"ability": "math",
"data_source": "aime",
"index": 798,
"split": "train"
} |
[
{
"content": "Question:\nIn acute triangle $ABC,$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\\overline{AB}$ and from $B$ to $\\overline{AC}$ , respectively. Line $PQ$ intersects the circumcircle of $\\triangle ABC$ in two distinct points, $X$ and $Y$ . Suppose $XP=10$ , $PQ=25$ , and $Q... | 574 | {
"ability": "math",
"data_source": "aime",
"index": 799,
"split": "train"
} |
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