context_messages listlengths 1 1 | ground_truth stringlengths 1 3 | meta dict |
|---|---|---|
[
{
"content": "Question:\nLet $\\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{abc}$ where $a, b, c$ are distinct digits. Find the sum of the elements of $\\mathcal{S}.$\nPlease reason step by step, and put your final answer within \\boxed{ }.",
... | 360 | {
"ability": "math",
"data_source": "aime",
"index": 400,
"split": "train"
} |
[
{
"content": "Question:\nAn angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $C$ to the area of shaded region $B$ is $\\frac{11}{5}$ . Find the ratio of shaded region $D$ to the area of shaded region $A$ . [asy] size(6cm); defaultpen(linewidth(0.7)+fontsi... | 408 | {
"ability": "math",
"data_source": "aime",
"index": 401,
"split": "train"
} |
[
{
"content": "Question:\nHexagon $ABCDEF$ is divided into five rhombuses, $P, Q, R, S,$ and $T$ , as shown. Rhombuses $P, Q, R,$ and $S$ are congruent, and each has area $\\sqrt{2006}.$ Let $K$ be the area of rhombus $T$ . Given that $K$ is a positive integer, find the number of possible values for $K.$ [asy] /... | 89 | {
"ability": "math",
"data_source": "aime",
"index": 402,
"split": "train"
} |
[
{
"content": "Question:\nThe sequence $a_1, a_2, \\ldots$ is geometric with $a_1=a$ and common ratio $r,$ where $a$ and $r$ are positive integers. Given that $\\log_8 a_1+\\log_8 a_2+\\cdots+\\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r).$\nPlease reason step by step, and put your fina... | 46 | {
"ability": "math",
"data_source": "aime",
"index": 403,
"split": "train"
} |
[
{
"content": "Question:\nEight circles of diameter 1 are packed in the first quadrant of the coordinate plane as shown. Let region $\\mathcal{R}$ be the union of the eight circular regions. Line $l,$ with slope 3, divides $\\mathcal{R}$ into two regions of equal area. Line $l$ 's equation can be expressed in th... | 65 | {
"ability": "math",
"data_source": "aime",
"index": 404,
"split": "train"
} |
[
{
"content": "Question:\nFind the sum of the values of $x$ such that $\\cos^3 3x+ \\cos^3 5x = 8 \\cos^3 4x \\cos^3 x,$ where $x$ is measured in degrees and $100< x< 200.$\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 906 | {
"ability": "math",
"data_source": "aime",
"index": 405,
"split": "train"
} |
[
{
"content": "Question:\nFor each even positive integer $x,$ let $g(x)$ denote the greatest power of 2 that divides $x.$ For example, $g(20)=4$ and $g(16)=16.$ For each positive integer $n,$ let $S_n=\\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than 1000 such that $S_n$ is a perfect square.\n... | 899 | {
"ability": "math",
"data_source": "aime",
"index": 406,
"split": "train"
} |
[
{
"content": "Question:\nA tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be th... | 183 | {
"ability": "math",
"data_source": "aime",
"index": 407,
"split": "train"
} |
[
{
"content": "Question:\nGiven that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\\ge 1,$ find the minimum possible value of $|x_1+x_2+\\cdots+x_{2006}|.$\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 27 | {
"ability": "math",
"data_source": "aime",
"index": 408,
"split": "train"
} |
[
{
"content": "Question:\nIn convex hexagon $ABCDEF$ , all six sides are congruent, $\\angle A$ and $\\angle D$ are right angles, and $\\angle B, \\angle C, \\angle E,$ and $\\angle F$ are congruent. The area of the hexagonal region is $2116(\\sqrt{2}+1).$ Find $AB$ .\nPlease reason step by step, and put your fi... | 46 | {
"ability": "math",
"data_source": "aime",
"index": 409,
"split": "train"
} |
[
{
"content": "Question:\nThe lengths of the sides of a triangle with positive area are $\\log_{10} 12$ , $\\log_{10} 75$ , and $\\log_{10} n$ , where $n$ is a positive integer. Find the number of possible values for $n$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "u... | 893 | {
"ability": "math",
"data_source": "aime",
"index": 410,
"split": "train"
} |
[
{
"content": "Question:\nLet $P$ be the product of the first 100 positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 49 | {
"ability": "math",
"data_source": "aime",
"index": 411,
"split": "train"
} |
[
{
"content": "Question:\nWhen rolling a certain unfair six-sided die with faces numbered $1, 2, 3, 4, 5$ , and $6$ , the probability of obtaining face $F$ is greater than $\\frac{1}{6}$ , the probability of obtaining the face opposite is less than $\\frac{1}{6}$ , the probability of obtaining any one of the oth... | 29 | {
"ability": "math",
"data_source": "aime",
"index": 412,
"split": "train"
} |
[
{
"content": "Question:\nSquare $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\\overline{BC}$ and $\\overline{CD},$ respectively, so that $\\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\\overline{AE}.$ The length of a side of ... | 12 | {
"ability": "math",
"data_source": "aime",
"index": 413,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit.\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 738 | {
"ability": "math",
"data_source": "aime",
"index": 414,
"split": "train"
} |
[
{
"content": "Question:\nThere is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distin... | 336 | {
"ability": "math",
"data_source": "aime",
"index": 415,
"split": "train"
} |
[
{
"content": "Question:\nCircles $\\mathcal{C}_1, \\mathcal{C}_2,$ and $\\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\\mathcal{C}_1$ and $\\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common... | 27 | {
"ability": "math",
"data_source": "aime",
"index": 416,
"split": "train"
} |
[
{
"content": "Question:\nSeven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 point... | 831 | {
"ability": "math",
"data_source": "aime",
"index": 417,
"split": "train"
} |
[
{
"content": "Question:\nA sequence is defined as follows $a_1=a_2=a_3=1,$ and, for all positive integers $n, a_{n+3}=a_{n+2}+a_{n+1}+a_n.$ Given that $a_{28}=6090307, a_{29}=11201821,$ and $a_{30}=20603361,$ find the remainder when $\\sum^{28}_{k=1} a_k$ is divided by 1000.\nPlease reason step by step, and put... | 834 | {
"ability": "math",
"data_source": "aime",
"index": 418,
"split": "train"
} |
[
{
"content": "Question:\nEquilateral $\\triangle ABC$ is inscribed in a circle of radius 2. Extend $\\overline{AB}$ through $B$ to point $D$ so that $AD=13,$ and extend $\\overline{AC}$ through $C$ to point $E$ so that $AE = 11.$ Through $D,$ draw a line $l_1$ parallel to $\\overline{AE},$ and through $E,$ draw... | 865 | {
"ability": "math",
"data_source": "aime",
"index": 419,
"split": "train"
} |
[
{
"content": "Question:\nHow many integers $N$ less than 1000 can be written as the sum of $j$ consecutive positive odd integers from exactly 5 values of $j\\ge 1$ ?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 15 | {
"ability": "math",
"data_source": "aime",
"index": 420,
"split": "train"
} |
[
{
"content": "Question:\nLet $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 63 | {
"ability": "math",
"data_source": "aime",
"index": 421,
"split": "train"
} |
[
{
"content": "Question:\nGiven that $x, y,$ and $z$ are real numbers that satisfy: \\begin{align*} x &= \\sqrt{y^2-\\frac{1}{16}}+\\sqrt{z^2-\\frac{1}{16}}, \\\\ y &= \\sqrt{z^2-\\frac{1}{25}}+\\sqrt{x^2-\\frac{1}{25}}, \\\\ z &= \\sqrt{x^2 - \\frac 1{36}}+\\sqrt{y^2-\\frac 1{36}}, \\end{align*} and that $x+y+z... | 9 | {
"ability": "math",
"data_source": "aime",
"index": 422,
"split": "train"
} |
[
{
"content": "Question:\nHow many positive perfect squares less than $10^6$ are multiples of 24?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 83 | {
"ability": "math",
"data_source": "aime",
"index": 423,
"split": "train"
} |
[
{
"content": "Question:\nA 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after ... | 52 | {
"ability": "math",
"data_source": "aime",
"index": 424,
"split": "train"
} |
[
{
"content": "Question:\nThe complex number $z$ is equal to $9+bi$ , where $b$ is a positive real number and $i^{2}=-1$ . Given that the imaginary parts of $z^{2}$ and $z^{3}$ are the same, what is $b$ equal to?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
... | 15 | {
"ability": "math",
"data_source": "aime",
"index": 425,
"split": "train"
} |
[
{
"content": "Question:\nThree planets orbit a star circularly in the same plane. Each moves in the same direction and moves at constant speed. Their periods are $60$ , $84$ , and $140$ years. The three planets and the star are currently collinear. What is the fewest number of years from now that they will ... | 105 | {
"ability": "math",
"data_source": "aime",
"index": 426,
"split": "train"
} |
[
{
"content": "Question:\nThe formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \\frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest int... | 539 | {
"ability": "math",
"data_source": "aime",
"index": 427,
"split": "train"
} |
[
{
"content": "Question:\nA frog is placed at the origin on the number line , and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of $3$ , or to the closest point with a greater integer coordinate that is a... | 169 | {
"ability": "math",
"data_source": "aime",
"index": 428,
"split": "train"
} |
[
{
"content": "Question:\nLet $N = \\sum_{k = 1}^{1000} k ( \\lceil \\log_{\\sqrt{2}} k \\rceil - \\lfloor \\log_{\\sqrt{2}} k \\rfloor )$ Find the remainder when $N$ is divided by 1000. ( $\\lfloor{k}\\rfloor$ is the greatest integer less than or equal to $k$ , and $\\lceil{k}\\rceil$ is the least integer gre... | 477 | {
"ability": "math",
"data_source": "aime",
"index": 429,
"split": "train"
} |
[
{
"content": "Question:\nThe polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_1(x) = x^2 + (k-29)x - k$ and $Q_2(x) = 2x^2+ (2k-43)x + k$ are both factors of $P(x)$ ?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 30 | {
"ability": "math",
"data_source": "aime",
"index": 430,
"split": "train"
} |
[
{
"content": "Question:\nIn right triangle $ABC$ with right angle $C$ , $CA = 30$ and $CB = 16$ . Its legs $CA$ and $CB$ are extended beyond $A$ and $B$ . Points $O_1$ and $O_2$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_1$ is tangent to... | 737 | {
"ability": "math",
"data_source": "aime",
"index": 431,
"split": "train"
} |
[
{
"content": "Question:\nIn a $6 \\times 4$ grid ( $6$ rows, $4$ columns), $12$ of the $24$ squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $N$ be the number of shadings with this property. Find the remainder when $N$ is divided by $1000$ ... | 860 | {
"ability": "math",
"data_source": "aime",
"index": 432,
"split": "train"
} |
[
{
"content": "Question:\nFor each positive integer $p$ , let $b(p)$ denote the unique positive integer $k$ such that $|k-\\sqrt{p}| < \\frac{1}{2}$ . For example, $b(6) = 2$ and $b(23) = 5$ . If $S = \\sum_{p=1}^{2007} b(p),$ find the remainder when $S$ is divided by 1000.\nPlease reason step by step, and put... | 955 | {
"ability": "math",
"data_source": "aime",
"index": 433,
"split": "train"
} |
[
{
"content": "Question:\nIn isosceles triangle $ABC$ , $A$ is located at the origin and $B$ is located at (20,0). Point $C$ is in the first quadrant with $AC = BC$ and angle $BAC = 75^{\\circ}$ . If triangle $ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive $y$ -axi... | 875 | {
"ability": "math",
"data_source": "aime",
"index": 434,
"split": "train"
} |
[
{
"content": "Question:\nA square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $AE$ , $BC$ , and $CD$ . The plane's intersection with the pyramid has an area that can be expressed as $\\sqrt{p}$ . Find $p$ .\nPlease reason step by step, and put ... | 80 | {
"ability": "math",
"data_source": "aime",
"index": 435,
"split": "train"
} |
[
{
"content": "Question:\nA sequence is defined over non-negative integral indexes in the following way: $a_{0}=a_{1}=3$ , $a_{n+1}a_{n-1}=a_{n}^{2}+2007$ . Find the greatest integer that does not exceed $\\frac{a_{2006}^{2}+a_{2007}^{2}}{a_{2006}a_{2007}}$\nPlease reason step by step, and put your final answer ... | 224 | {
"ability": "math",
"data_source": "aime",
"index": 436,
"split": "train"
} |
[
{
"content": "Question:\nLet $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$ , respectively, with $FA = 5$ and $CD = 2$ . Point $E$ lies on side $CA$ such that angle $DEF = 60^{\\circ}$ . The area of triangle $DEF$ is $14\\sqrt{3}$ . The two possible values of the lengt... | 989 | {
"ability": "math",
"data_source": "aime",
"index": 437,
"split": "train"
} |
[
{
"content": "Question:\nA mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in $2007$ . No character may appear in a sequence more times than it appears among the four letters ... | 372 | {
"ability": "math",
"data_source": "aime",
"index": 438,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of ordered triples $(a,b,c)$ where $a$ , $b$ , and $c$ are positive integers , $a$ is a factor of $b$ , $a$ is a factor of $c$ , and $a+b+c=100$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 200 | {
"ability": "math",
"data_source": "aime",
"index": 439,
"split": "train"
} |
[
{
"content": "Question:\nThe workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, $100$ workers can produce $300$ widgets and $200$ whoosits. In two hours, $60$ workers can pro... | 450 | {
"ability": "math",
"data_source": "aime",
"index": 440,
"split": "train"
} |
[
{
"content": "Question:\nThe graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant ?\nPlease reason step by step, and p... | 888 | {
"ability": "math",
"data_source": "aime",
"index": 441,
"split": "train"
} |
[
{
"content": "Question:\nAn integer is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd , and $a_{i}>a_{i+1}$ if $a_{i}$ is even . How many four-digit parity-monotonic integers are there?\nPlease reason step by step, and put your fi... | 640 | {
"ability": "math",
"data_source": "aime",
"index": 442,
"split": "train"
} |
[
{
"content": "Question:\nGiven a real number $x,$ let $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \\ldots, n_{70}$ such that $k=\\lfloor\\sqrt[3]{n_{1}}\\rfloor = \\lfloor\\sqrt[3]{n_{2}}\\rfloor ... | 553 | {
"ability": "math",
"data_source": "aime",
"index": 443,
"split": "train"
} |
[
{
"content": "Question:\nRectangle $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The inscribed circle of triangle $BEF$ is tangent to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at point $Q.$ Find $PQ.$... | 259 | {
"ability": "math",
"data_source": "aime",
"index": 444,
"split": "train"
} |
[
{
"content": "Question:\nLet $S$ be a set with six elements . Let $\\mathcal{P}$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$ , not necessarily distinct, are chosen independently and at random from $\\mathcal{P}$ . The probability that $B$ is contained in one of $A$ or $S-A$ is $\\frac{m}{n^{r}}... | 710 | {
"ability": "math",
"data_source": "aime",
"index": 445,
"split": "train"
} |
[
{
"content": "Question:\nTwo long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface . The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$ . It rolls over the smaller tube and continues rolling along th... | 179 | {
"ability": "math",
"data_source": "aime",
"index": 446,
"split": "train"
} |
[
{
"content": "Question:\nThe increasing geometric sequence $x_{0},x_{1},x_{2},\\ldots$ consists entirely of integral powers of $3.$ Given that $\\sum_{n=0}^{7}\\log_{3}(x_{n}) = 308$ and $56 \\leq \\log_{3}\\left ( \\sum_{n=0}^{7}x_{n}\\right ) \\leq 57,$ find $\\log_{3}(x_{14}).$\nPlease reason step by step, a... | 91 | {
"ability": "math",
"data_source": "aime",
"index": 447,
"split": "train"
} |
[
{
"content": "Question:\nA triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$ th row for $1 \\leq k \\leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). I... | 640 | {
"ability": "math",
"data_source": "aime",
"index": 448,
"split": "train"
} |
[
{
"content": "Question:\nLet $f(x)$ 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 $f(5).$\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 676 | {
"ability": "math",
"data_source": "aime",
"index": 449,
"split": "train"
} |
[
{
"content": "Question:\nFour circles $\\omega,$ $\\omega_{A},$ $\\omega_{B},$ and $\\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\\omega_{A}$ is tangent to sides $AB$ and $AC$ , $\\omega_{B}$ to $BC$ and $BA$ , $\\omega_{C}$ to $CA$ and $CB$ , and $\\omega$ is externa... | 389 | {
"ability": "math",
"data_source": "aime",
"index": 450,
"split": "train"
} |
[
{
"content": "Question:\nOf the students attending a school party, $60\\%$ of the students are girls, and $40\\%$ of the students like to dance. After these students are joined by $20$ more boy students, all of whom like to dance, the party is now $58\\%$ girls. How many students now at the party like to dance?... | 252 | {
"ability": "math",
"data_source": "aime",
"index": 451,
"split": "train"
} |
[
{
"content": "Question:\nSquare $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$ , and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\\triangle GEM$ .\nPlease reason step by step, and put your final answer wit... | 25 | {
"ability": "math",
"data_source": "aime",
"index": 452,
"split": "train"
} |
[
{
"content": "Question:\nEd and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $74$ kilometers after biking for $2$ hours, jogging for $3$ hours, and swimming for $4$ hours, while Sue covers $91$ kilometers after jogg... | 314 | {
"ability": "math",
"data_source": "aime",
"index": 453,
"split": "train"
} |
[
{
"content": "Question:\nThere exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$ . Find $x + y$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 80 | {
"ability": "math",
"data_source": "aime",
"index": 454,
"split": "train"
} |
[
{
"content": "Question:\nA right circular cone has base radius $r$ and height $h$ . The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches t... | 14 | {
"ability": "math",
"data_source": "aime",
"index": 455,
"split": "train"
} |
[
{
"content": "Question:\nA triangular array of numbers has a first row consisting of the odd integers $1,3,5,\\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the... | 17 | {
"ability": "math",
"data_source": "aime",
"index": 456,
"split": "train"
} |
[
{
"content": "Question:\nLet $S_i$ be the set of all integers $n$ such that $100i\\leq n < 100(i + 1)$ . For example, $S_4$ is the set ${400,401,402,\\ldots,499}$ . How many of the sets $S_0, S_1, S_2, \\ldots, S_{999}$ do not contain a perfect square?\nPlease reason step by step, and put your final answer wi... | 708 | {
"ability": "math",
"data_source": "aime",
"index": 457,
"split": "train"
} |
[
{
"content": "Question:\nFind the positive integer $n$ such that \\[\\arctan\\frac {1}{3} + \\arctan\\frac {1}{4} + \\arctan\\frac {1}{5} + \\arctan\\frac {1}{n} = \\frac {\\pi}{4}.\\]\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 47 | {
"ability": "math",
"data_source": "aime",
"index": 458,
"split": "train"
} |
[
{
"content": "Question:\nTen identical crates each of dimensions $3$ ft $\\times$ $4$ ft $\\times$ $6$ ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $\\frac {... | 190 | {
"ability": "math",
"data_source": "aime",
"index": 459,
"split": "train"
} |
[
{
"content": "Question:\nLet $ABCD$ be an isosceles trapezoid with $\\overline{AD}||\\overline{BC}$ whose angle at the longer base $\\overline{AD}$ is $\\dfrac{\\pi}{3}$ . The diagonals have length $10\\sqrt {21}$ , and point $E$ is at distances $10\\sqrt {7}$ and $30\\sqrt {7}$ from vertices $A$ and $D$ , resp... | 32 | {
"ability": "math",
"data_source": "aime",
"index": 460,
"split": "train"
} |
[
{
"content": "Question:\nConsider sequences that consist entirely of $A$ 's and $B$ 's and that have the property that every run of consecutive $A$ 's has even length, and every run of consecutive $B$ 's has odd length. Examples of such sequences are $AA$ , $B$ , and $AABAA$ , while $BBAB$ is not such a sequenc... | 172 | {
"ability": "math",
"data_source": "aime",
"index": 461,
"split": "train"
} |
[
{
"content": "Question:\nOn a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (T... | 375 | {
"ability": "math",
"data_source": "aime",
"index": 462,
"split": "train"
} |
[
{
"content": "Question:\nLet $p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3$ . Suppose that $p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1) = p(1,1) = p(1, - 1) = p(2,2) = 0$ . There is a point $\\left(\\frac {a}{c},\\frac {b}{c}\\right)$ for which $p\\left(\\frac ... | 40 | {
"ability": "math",
"data_source": "aime",
"index": 463,
"split": "train"
} |
[
{
"content": "Question:\nLet $\\overline{AB}$ be a diameter of circle $\\omega$ . Extend $\\overline{AB}$ through $A$ to $C$ . Point $T$ lies on $\\omega$ so that line $CT$ is tangent to $\\omega$ . Point $P$ is the foot of the perpendicular from $A$ to line $CT$ . Suppose $\\overline{AB} = 18$ , and let $m$ de... | 432 | {
"ability": "math",
"data_source": "aime",
"index": 464,
"split": "train"
} |
[
{
"content": "Question:\nLet $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \\cdots + 4^2 + 3^2 - 2^2 - 1^2$ , where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by $1000$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
... | 100 | {
"ability": "math",
"data_source": "aime",
"index": 465,
"split": "train"
} |
[
{
"content": "Question:\nRudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at ... | 620 | {
"ability": "math",
"data_source": "aime",
"index": 466,
"split": "train"
} |
[
{
"content": "Question:\nA block of cheese in the shape of a rectangular solid measures $10$ cm by $13$ cm by $14$ cm. Ten slices are cut from the cheese. Each slice has a width of $1$ cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the... | 729 | {
"ability": "math",
"data_source": "aime",
"index": 467,
"split": "train"
} |
[
{
"content": "Question:\nThere exist $r$ unique nonnegative integers $n_1 > n_2 > \\cdots > n_r$ and $r$ integers $a_k$ ( $1\\le k\\le r$ ) with each $a_k$ either $1$ or $- 1$ such that \\[a_13^{n_1} + a_23^{n_2} + \\cdots + a_r3^{n_r} = 2008.\\] Find $n_1 + n_2 + \\cdots + n_r$ .\nPlease reason step by step, a... | 21 | {
"ability": "math",
"data_source": "aime",
"index": 468,
"split": "train"
} |
[
{
"content": "Question:\nIn trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$ , let $BC = 1000$ and $AD = 2008$ . Let $\\angle A = 37^\\circ$ , $\\angle D = 53^\\circ$ , and $M$ and $N$ be the midpoints of $\\overline{BC}$ and $\\overline{AD}$ , respectively. Find the length $MN$ .\nPlease reason s... | 504 | {
"ability": "math",
"data_source": "aime",
"index": 469,
"split": "train"
} |
[
{
"content": "Question:\nThe sequence $\\{a_n\\}$ is defined by \\[a_0 = 1,a_1 = 1, \\text{ and } a_n = a_{n - 1} + \\frac {a_{n - 1}^2}{a_{n - 2}}\\text{ for }n\\ge2.\\] The sequence $\\{b_n\\}$ is defined by \\[b_0 = 1,b_1 = 3, \\text{ and } b_n = b_{n - 1} + \\frac {b_{n - 1}^2}{b_{n - 2}}\\text{ for }n\\ge2... | 561 | {
"ability": "math",
"data_source": "aime",
"index": 470,
"split": "train"
} |
[
{
"content": "Question:\nLet $r$ , $s$ , and $t$ be the three roots of the equation \\[8x^3 + 1001x + 2008 = 0.\\] Find $(r + s)^3 + (s + t)^3 + (t + r)^3$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 753 | {
"ability": "math",
"data_source": "aime",
"index": 471,
"split": "train"
} |
[
{
"content": "Question:\nLet $a = \\pi/2008$ . Find the smallest positive integer $n$ such that \\[2[\\cos(a)\\sin(a) + \\cos(4a)\\sin(2a) + \\cos(9a)\\sin(3a) + \\cdots + \\cos(n^2a)\\sin(na)]\\] is an integer.\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 251 | {
"ability": "math",
"data_source": "aime",
"index": 472,
"split": "train"
} |
[
{
"content": "Question:\nA particle is located on the coordinate plane at $(5,0)$ . Define a move for the particle as a counterclockwise rotation of $\\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$ -direction. Given that the particle's position after $150$ moves is $... | 19 | {
"ability": "math",
"data_source": "aime",
"index": 473,
"split": "train"
} |
[
{
"content": "Question:\nIn triangle $ABC$ , $AB = AC = 100$ , and $BC = 56$ . Circle $P$ has radius $16$ and is tangent to $\\overline{AC}$ and $\\overline{BC}$ . Circle $Q$ is externally tangent to circle $P$ and is tangent to $\\overline{AB}$ and $\\overline{BC}$ . No point of circle $Q$ lies outside of $\\b... | 254 | {
"ability": "math",
"data_source": "aime",
"index": 474,
"split": "train"
} |
[
{
"content": "Question:\nThere are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags ... | 310 | {
"ability": "math",
"data_source": "aime",
"index": 475,
"split": "train"
} |
[
{
"content": "Question:\nA regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \\left\\lbrace\\frac{1}{z}|z \\in R\\right\\rbrace$ . Then the area o... | 29 | {
"ability": "math",
"data_source": "aime",
"index": 476,
"split": "train"
} |
[
{
"content": "Question:\nLet $a$ and $b$ be positive real numbers with $a \\ge b$ . Let $\\rho$ be the maximum possible value of $\\dfrac{a}{b}$ for which the system of equations \\[a^2 + y^2 = b^2 + x^2 = (a-x)^2 + (b-y)^2\\] has a solution $(x,y)$ satisfying $0 \\le x < a$ and $0 \\le y < b$ . Then $\\rho^2$ ... | 7 | {
"ability": "math",
"data_source": "aime",
"index": 477,
"split": "train"
} |
[
{
"content": "Question:\nCall a $3$ -digit number geometric if it has $3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role... | 840 | {
"ability": "math",
"data_source": "aime",
"index": 478,
"split": "train"
} |
[
{
"content": "Question:\nThere is a complex number $z$ with imaginary part $164$ and a positive integer $n$ such that \\[\\frac {z}{z + n} = 4i.\\] Find $n$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 697 | {
"ability": "math",
"data_source": "aime",
"index": 479,
"split": "train"
} |
[
{
"content": "Question:\nA coin that comes up heads with probability $p > 0$ and tails with probability $1 - p > 0$ independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to $\\frac {1}{25}$ of the probability of five heads and three tails. Let $p = \\f... | 11 | {
"ability": "math",
"data_source": "aime",
"index": 480,
"split": "train"
} |
[
{
"content": "Question:\nIn parallelogram $ABCD$ , point $M$ is on $\\overline{AB}$ so that $\\frac {AM}{AB} = \\frac {17}{1000}$ and point $N$ is on $\\overline{AD}$ so that $\\frac {AN}{AD} = \\frac {17}{2009}$ . Let $P$ be the point of intersection of $\\overline{AC}$ and $\\overline{MN}$ . Find $\\frac {AC}... | 177 | {
"ability": "math",
"data_source": "aime",
"index": 481,
"split": "train"
} |
[
{
"content": "Question:\nTriangle $ABC$ has $AC = 450$ and $BC = 300$ . Points $K$ and $L$ are located on $\\overline{AC}$ and $\\overline{AB}$ respectively so that $AK = CK$ , and $\\overline{CL}$ is the angle bisector of angle $C$ . Let $P$ be the point of intersection of $\\overline{BK}$ and $\\overline{CL}$... | 72 | {
"ability": "math",
"data_source": "aime",
"index": 482,
"split": "train"
} |
[
{
"content": "Question:\nHow many positive integers $N$ less than $1000$ are there such that the equation $x^{\\lfloor x\\rfloor} = N$ has a solution for $x$ ?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 412 | {
"ability": "math",
"data_source": "aime",
"index": 483,
"split": "train"
} |
[
{
"content": "Question:\nThe sequence $(a_n)$ satisfies $a_1 = 1$ and $5^{(a_{n + 1} - a_n)} - 1 = \\frac {1}{n + \\frac {2}{3}}$ for $n \\geq 1$ . Let $k$ be the least integer greater than $1$ for which $a_k$ is an integer. Find $k$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
... | 41 | {
"ability": "math",
"data_source": "aime",
"index": 484,
"split": "train"
} |
[
{
"content": "Question:\nLet $S = \\{2^0,2^1,2^2,\\ldots,2^{10}\\}$ . Consider all possible positive differences of pairs of elements of $S$ . Let $N$ be the sum of all of these differences. Find the remainder when $N$ is divided by $1000$ .\nPlease reason step by step, and put your final answer within \\boxed{... | 398 | {
"ability": "math",
"data_source": "aime",
"index": 485,
"split": "train"
} |
[
{
"content": "Question:\nA game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $\\text{\\textdollar}1$ to $\\text{\\textdollar}9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, t... | 420 | {
"ability": "math",
"data_source": "aime",
"index": 486,
"split": "train"
} |
[
{
"content": "Question:\nThe Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15$ in clockwise order. Committee rules state that a Martian ... | 346 | {
"ability": "math",
"data_source": "aime",
"index": 487,
"split": "train"
} |
[
{
"content": "Question:\nConsider 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.\nPlease reason step by ... | 600 | {
"ability": "math",
"data_source": "aime",
"index": 488,
"split": "train"
} |
[
{
"content": "Question:\nIn right $\\triangle ABC$ with hypotenuse $\\overline{AB}$ , $AC = 12$ , $BC = 35$ , and $\\overline{CD}$ is the altitude to $\\overline{AB}$ . Let $\\omega$ be the circle having $\\overline{CD}$ as a diameter. Let $I$ be a point outside $\\triangle ABC$ such that $\\overline{AI}$ and $... | 11 | {
"ability": "math",
"data_source": "aime",
"index": 489,
"split": "train"
} |
[
{
"content": "Question:\nThe terms of the sequence $\\{a_i\\}$ defined by $a_{n + 2} = \\frac {a_n + 2009} {1 + a_{n + 1}}$ for $n \\ge 1$ are positive integers. Find the minimum possible value of $a_1 + a_2$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 90 | {
"ability": "math",
"data_source": "aime",
"index": 490,
"split": "train"
} |
[
{
"content": "Question:\nFor $t = 1, 2, 3, 4$ , define $S_t = \\sum_{i = 1}^{350}a_i^t$ , where $a_i \\in \\{1,2,3,4\\}$ . If $S_1 = 513$ and $S_4 = 4745$ , find the minimum possible value for $S_2$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 905 | {
"ability": "math",
"data_source": "aime",
"index": 491,
"split": "train"
} |
[
{
"content": "Question:\nIn triangle $ABC$ , $AB = 10$ , $BC = 14$ , and $CA = 16$ . Let $D$ be a point in the interior of $\\overline{BC}$ . Let points $I_B$ and $I_C$ denote the incenters of triangles $ABD$ and $ACD$ , respectively. The circumcircles of triangles $BI_BD$ and $CI_CD$ meet at distinct points $P... | 150 | {
"ability": "math",
"data_source": "aime",
"index": 492,
"split": "train"
} |
[
{
"content": "Question:\nBefore 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 ne... | 114 | {
"ability": "math",
"data_source": "aime",
"index": 493,
"split": "train"
} |
[
{
"content": "Question:\nSuppose that $a$ , $b$ , and $c$ are positive real numbers such that $a^{\\log_3 7} = 27$ , $b^{\\log_7 11} = 49$ , and $c^{\\log_{11}25} = \\sqrt{11}$ . Find \\[a^{(\\log_3 7)^2} + b^{(\\log_7 11)^2} + c^{(\\log_{11} 25)^2}.\\]\nPlease reason step by step, and put your final answer wit... | 469 | {
"ability": "math",
"data_source": "aime",
"index": 494,
"split": "train"
} |
[
{
"content": "Question:\nIn rectangle $ABCD$ , $AB=100$ . Let $E$ be the midpoint of $\\overline{AD}$ . Given that line $AC$ and line $BE$ are perpendicular, find the greatest integer less than $AD$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 141 | {
"ability": "math",
"data_source": "aime",
"index": 495,
"split": "train"
} |
[
{
"content": "Question:\nA group of children held a grape-eating contest. When the contest was over, the winner had eaten $n$ grapes, and the child in $k$ -th place had eaten $n+2-2k$ grapes. The total number of grapes eaten in the contest was $2009$ . Find the smallest possible value of $n$ .\nPlease reason st... | 89 | {
"ability": "math",
"data_source": "aime",
"index": 496,
"split": "train"
} |
[
{
"content": "Question:\nEquilateral triangle $T$ is inscribed in circle $A$ , which has radius $10$ . Circle $B$ with radius $3$ is internally tangent to circle $A$ at one vertex of $T$ . Circles $C$ and $D$ , both with radius $2$ , are internally tangent to circle $A$ at the other two vertices of $T$ . Circle... | 32 | {
"ability": "math",
"data_source": "aime",
"index": 497,
"split": "train"
} |
[
{
"content": "Question:\nLet $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$ .\nPlease reason step by step, and put your final answer within \\bo... | 750 | {
"ability": "math",
"data_source": "aime",
"index": 498,
"split": "train"
} |
[
{
"content": "Question:\nDefine $n!!$ to be $n(n-2)(n-4)\\cdots 3\\cdot 1$ for $n$ odd and $n(n-2)(n-4)\\cdots 4\\cdot 2$ for $n$ even. When $\\sum_{i=1}^{2009} \\frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\\dfrac{ab}{10}$ .\nPlease reason s... | 401 | {
"ability": "math",
"data_source": "aime",
"index": 499,
"split": "train"
} |
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