context_messages listlengths 1 1 | ground_truth stringlengths 1 3 | meta dict |
|---|---|---|
[
{
"content": "Question:\nLet $T = \\{9^k : k ~ \\mbox{is an integer}, 0 \\le k \\le 4000\\}$ . Given that $9^{4000}_{}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T_{}^{}$ have 9 as their leftmost digit?\nPlease reason step by step, and put your final answer within \\boxed{... | 184 | {
"ability": "math",
"data_source": "aime",
"index": 100,
"split": "train"
} |
[
{
"content": "Question:\nThe rectangle $ABCD^{}_{}$ below has dimensions $AB^{}_{} = 12 \\sqrt{3}$ and $BC^{}_{} = 13 \\sqrt{3}$ . Diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect at $P^{}_{}$ . If triangle $ABP^{}_{}$ is cut out and removed, edges $\\overline{AP}$ and $\\overline{BP}$ are joined, an... | 594 | {
"ability": "math",
"data_source": "aime",
"index": 101,
"split": "train"
} |
[
{
"content": "Question:\nFind $ax^5 + by^5$ if the real numbers $a,b,x,$ and $y$ satisfy the equations \\begin{align*} ax + by &= 3, \\\\ ax^2 + by^2 &= 7, \\\\ ax^3 + by^3 &= 16, \\\\ ax^4 + by^4 &= 42. \\end{align*}\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"... | 20 | {
"ability": "math",
"data_source": "aime",
"index": 102,
"split": "train"
} |
[
{
"content": "Question:\nFind $x^2+y^2_{}$ if $x_{}^{}$ and $y_{}^{}$ are positive integers such that \\begin{align*} xy+x+y&=71, \\\\ x^2y+xy^2&=880. \\end{align*}\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 146 | {
"ability": "math",
"data_source": "aime",
"index": 103,
"split": "train"
} |
[
{
"content": "Question:\nRectangle $ABCD_{}^{}$ has sides $\\overline {AB}$ of length 4 and $\\overline {CB}$ of length 3. Divide $\\overline {AB}$ into 168 congruent segments with points $A_{}^{}=P_0, P_1, \\ldots, P_{168}=B$ , and divide $\\overline {CB}$ into 168 congruent segments with points $C_{}^{}=Q_0, ... | 840 | {
"ability": "math",
"data_source": "aime",
"index": 104,
"split": "train"
} |
[
{
"content": "Question:\nHow many real numbers $x^{}_{}$ satisfy the equation $\\frac{1}{5}\\log_2 x = \\sin (5\\pi x)$ ?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 159 | {
"ability": "math",
"data_source": "aime",
"index": 105,
"split": "train"
} |
[
{
"content": "Question:\nGiven a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?\nPlease reason step by step, and put your final answer within \\... | 128 | {
"ability": "math",
"data_source": "aime",
"index": 106,
"split": "train"
} |
[
{
"content": "Question:\nFor how many real numbers $a^{}_{}$ does the quadratic equation $x^2 + ax^{}_{} + 6a=0$ have only integer roots for $x^{}_{}$ ?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 10 | {
"ability": "math",
"data_source": "aime",
"index": 107,
"split": "train"
} |
[
{
"content": "Question:\nSuppose that $\\sec x+\\tan x=\\frac{22}7$ and that $\\csc x+\\cot x=\\frac mn,$ where $\\frac mn$ is in lowest terms. Find $m+n^{}_{}.$\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 44 | {
"ability": "math",
"data_source": "aime",
"index": 108,
"split": "train"
} |
[
{
"content": "Question:\nTwo three-letter strings, $aaa^{}_{}$ and $bbb^{}_{}$ , are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a^{}_{}$ when it should have been a $b^{}_{}$ , or as a $... | 532 | {
"ability": "math",
"data_source": "aime",
"index": 109,
"split": "train"
} |
[
{
"content": "Question:\nTwelve 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. ... | 135 | {
"ability": "math",
"data_source": "aime",
"index": 110,
"split": "train"
} |
[
{
"content": "Question:\nRhombus $PQRS^{}_{}$ is inscribed in rectangle $ABCD^{}_{}$ so that vertices $P^{}_{}$ , $Q^{}_{}$ , $R^{}_{}$ , and $S^{}_{}$ are interior points on sides $\\overline{AB}$ , $\\overline{BC}$ , $\\overline{CD}$ , and $\\overline{DA}$ , respectively. It is given that $PB^{}_{}=15$ , $BQ^... | 677 | {
"ability": "math",
"data_source": "aime",
"index": 111,
"split": "train"
} |
[
{
"content": "Question:\nA drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\\frac{1}{2}$ that both are red or both are blue. What is the largest possible number of red soc... | 990 | {
"ability": "math",
"data_source": "aime",
"index": 112,
"split": "train"
} |
[
{
"content": "Question:\nA hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\\overline{AB}$ , has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$ .\nPlease reason step by step, and put your final answer within \\boxed{ }... | 384 | {
"ability": "math",
"data_source": "aime",
"index": 113,
"split": "train"
} |
[
{
"content": "Question:\nFind the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms.\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 400 | {
"ability": "math",
"data_source": "aime",
"index": 114,
"split": "train"
} |
[
{
"content": "Question:\nA positive integer is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"ro... | 502 | {
"ability": "math",
"data_source": "aime",
"index": 115,
"split": "train"
} |
[
{
"content": "Question:\nA tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly $0.500$ . During the weekend, she plays four matches, winning three and losing one. At the end of the w... | 164 | {
"ability": "math",
"data_source": "aime",
"index": 116,
"split": "train"
} |
[
{
"content": "Question:\nIn Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below. \\[\\begin{array}{c@{\\hspace{8em}} c@{\\hspace{6pt}}c@{\\hspace{6pt}}c@{\\hspace{6pt}}c@{\\hspace{4pt}}c@{\\hspace{2pt}} c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\h... | 62 | {
"ability": "math",
"data_source": "aime",
"index": 117,
"split": "train"
} |
[
{
"content": "Question:\nLet $S^{}_{}$ be the set of all rational numbers $r^{}_{}$ , $0^{}_{}<r<1$ , that have a repeating decimal expansion in the form $0.abcabcabc\\ldots=0.\\overline{abc}$ , where the digits $a^{}_{}$ , $b^{}_{}$ , and $c^{}_{}$ are not necessarily distinct. To write the elements of $S^{}_{... | 660 | {
"ability": "math",
"data_source": "aime",
"index": 118,
"split": "train"
} |
[
{
"content": "Question:\nFor how many pairs of consecutive integers in $\\{1000,1001,1002^{}_{},\\ldots,2000\\}$ is no carrying required when the two integers are added?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 156 | {
"ability": "math",
"data_source": "aime",
"index": 119,
"split": "train"
} |
[
{
"content": "Question:\nFaces $ABC^{}_{}$ and $BCD^{}_{}$ of tetrahedron $ABCD^{}_{}$ meet at an angle of $30^\\circ$ . The area of face $ABC^{}_{}$ is $120^{}_{}$ , the area of face $BCD^{}_{}$ is $80^{}_{}$ , and $BC=10^{}_{}$ . Find the volume of the tetrahedron.\nPlease reason step by step, and put your fi... | 320 | {
"ability": "math",
"data_source": "aime",
"index": 120,
"split": "train"
} |
[
{
"content": "Question:\nFor any sequence of real numbers $A=(a_1,a_2,a_3,\\ldots)$ , define $\\Delta A^{}_{}$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\\ldots)$ , whose $n^{th}$ term is $a_{n+1}-a_n^{}$ . Suppose that all of the terms of the sequence $\\Delta(\\Delta A^{}_{})$ are $1^{}_{}$ , and that $a_{... | 819 | {
"ability": "math",
"data_source": "aime",
"index": 121,
"split": "train"
} |
[
{
"content": "Question:\nTrapezoid $ABCD^{}_{}$ has sides $AB=92^{}_{}$ , $BC=50^{}_{}$ , $CD=19^{}_{}$ , and $AD=70^{}_{}$ , with $AB^{}_{}$ parallel to $CD^{}_{}$ . A circle with center $P^{}_{}$ on $AB^{}_{}$ is drawn tangent to $BC^{}_{}$ and $AD^{}_{}$ . Given that $AP^{}_{}=\\frac mn$ , where $m^{}_{}$ an... | 164 | {
"ability": "math",
"data_source": "aime",
"index": 122,
"split": "train"
} |
[
{
"content": "Question:\nConsider the region $A^{}_{}$ in the complex plane that consists of all points $z^{}_{}$ such that both $\\frac{z^{}_{}}{40}$ and $\\frac{40^{}_{}}{\\overline{z}}$ have real and imaginary parts between $0^{}_{}$ and $1^{}_{}$ , inclusive. What is the integer that is nearest the area of ... | 572 | {
"ability": "math",
"data_source": "aime",
"index": 123,
"split": "train"
} |
[
{
"content": "Question:\nLines $l_1^{}$ and $l_2^{}$ both pass through the origin and make first-quadrant angles of $\\frac{\\pi}{70}$ and $\\frac{\\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l^{}_{}$ , the transformation $R(l)^{}_{}$ produces another line as follows: $l^{}_{}$ is r... | 945 | {
"ability": "math",
"data_source": "aime",
"index": 124,
"split": "train"
} |
[
{
"content": "Question:\nIn a game of Chomp , two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes (\"eats\") all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightwar... | 792 | {
"ability": "math",
"data_source": "aime",
"index": 125,
"split": "train"
} |
[
{
"content": "Question:\nTriangle $ABC^{}_{}$ has $AB=9^{}_{}$ and $BC: AC=40: 41^{}_{}$ . What's the largest area that this triangle can have?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 820 | {
"ability": "math",
"data_source": "aime",
"index": 126,
"split": "train"
} |
[
{
"content": "Question:\nIn triangle $ABC^{}_{}$ , $A'$ , $B'$ , and $C'$ are on the sides $BC$ , $AC^{}_{}$ , and $AB^{}_{}$ , respectively. Given that $AA'$ , $BB'$ , and $CC'$ are concurrent at the point $O^{}_{}$ , and that $\\frac{AO^{}_{}}{OA'}+\\frac{BO}{OB'}+\\frac{CO}{OC'}=92$ , find $\\frac{AO}{OA'}\\... | 94 | {
"ability": "math",
"data_source": "aime",
"index": 127,
"split": "train"
} |
[
{
"content": "Question:\nDefine a positive integer $n^{}_{}$ to be a factorial tail if there is some positive integer $m^{}_{}$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails?\nPlease reason step by step, and put you... | 396 | {
"ability": "math",
"data_source": "aime",
"index": 128,
"split": "train"
} |
[
{
"content": "Question:\nHow many even integers between 4000 and 7000 have four different digits?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 728 | {
"ability": "math",
"data_source": "aime",
"index": 129,
"split": "train"
} |
[
{
"content": "Question:\nDuring a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $\\fra... | 580 | {
"ability": "math",
"data_source": "aime",
"index": 130,
"split": "train"
} |
[
{
"content": "Question:\nHow many ordered four-tuples of integers $(a,b,c,d)\\,$ with $0 < a < b < c < d < 500\\,$ satisfy $a + d = b + c\\,$ and $bc - ad = 93\\,$ ?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 870 | {
"ability": "math",
"data_source": "aime",
"index": 131,
"split": "train"
} |
[
{
"content": "Question:\nLet $P_0(x) = x^3 + 313x^2 - 77x - 8\\,$ . For integers $n \\ge 1\\,$ , define $P_n(x) = P_{n - 1}(x - n)\\,$ . What is the coefficient of $x\\,$ in $P_{20}(x)\\,$ ?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 763 | {
"ability": "math",
"data_source": "aime",
"index": 132,
"split": "train"
} |
[
{
"content": "Question:\nWhat is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 495 | {
"ability": "math",
"data_source": "aime",
"index": 133,
"split": "train"
} |
[
{
"content": "Question:\nThree numbers, $a_1, a_2, a_3$ , are drawn randomly and without replacement from the set $\\{1, 2, 3,\\ldots, 1000\\}$ . Three other numbers, $b_1, b_2, b_3$ , are then drawn randomly and without replacement from the remaining set of $997$ numbers. Let $p$ be the probability that, after... | 5 | {
"ability": "math",
"data_source": "aime",
"index": 134,
"split": "train"
} |
[
{
"content": "Question:\nLet $S\\,$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S\\,$ so that the union of the two subsets is $S\\,$ ? The order of selection does not matter; for example, the pair of subsets $\\{a, c\\},\\{b, c, d, e, f\\}$ rep... | 365 | {
"ability": "math",
"data_source": "aime",
"index": 135,
"split": "train"
} |
[
{
"content": "Question:\nTwo thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. (See figure.) Continue this process until ... | 118 | {
"ability": "math",
"data_source": "aime",
"index": 136,
"split": "train"
} |
[
{
"content": "Question:\nEuler's formula states that for a convex polyhedron with $V\\,$ vertices, $E\\,$ edges, and $F\\,$ faces, $V-E+F=2\\,$ . A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V\\,$ vertices, $T\\,$ triangular faces and $P^{}_{}$ p... | 250 | {
"ability": "math",
"data_source": "aime",
"index": 137,
"split": "train"
} |
[
{
"content": "Question:\nAlfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first... | 93 | {
"ability": "math",
"data_source": "aime",
"index": 138,
"split": "train"
} |
[
{
"content": "Question:\nThe vertices of $\\triangle ABC$ are $A = (0,0)\\,$ , $B = (0,420)\\,$ , and $C = (560,0)\\,$ . The six faces of a die are labeled with two $A\\,$ 's, two $B\\,$ 's, and two $C\\,$ 's. Point $P_1 = (k,m)\\,$ is chosen in the interior of $\\triangle ABC$ , and points $P_2\\,$ , $P_3\\,... | 344 | {
"ability": "math",
"data_source": "aime",
"index": 139,
"split": "train"
} |
[
{
"content": "Question:\nJenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the ... | 163 | {
"ability": "math",
"data_source": "aime",
"index": 140,
"split": "train"
} |
[
{
"content": "Question:\nA rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 r... | 448 | {
"ability": "math",
"data_source": "aime",
"index": 141,
"split": "train"
} |
[
{
"content": "Question:\nLet $\\overline{CH}$ be an altitude of $\\triangle ABC$ . Let $R\\,$ and $S\\,$ be the points where the circles inscribed in the triangles $ACH\\,$ and $BCH^{}_{}$ are tangent to $\\overline{CH}$ . If $AB = 1995\\,$ , $AC = 1994\\,$ , and $BC = 1993\\,$ , then $RS\\,$ can be expressed a... | 997 | {
"ability": "math",
"data_source": "aime",
"index": 142,
"split": "train"
} |
[
{
"content": "Question:\nThe increasing sequence $3, 15, 24, 48, \\ldots\\,$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000?\nPlease reason step by step, and put your final answer within \\boxed{ }... | 63 | {
"ability": "math",
"data_source": "aime",
"index": 143,
"split": "train"
} |
[
{
"content": "Question:\nA circle with diameter $\\overline{PQ}\\,$ of length 10 is internally tangent at $P^{}_{}$ to a circle of radius 20. Square $ABCD\\,$ is constructed with $A\\,$ and $B\\,$ on the larger circle, $\\overline{CD}\\,$ tangent at $Q\\,$ to the smaller circle, and the smaller circle outside $... | 312 | {
"ability": "math",
"data_source": "aime",
"index": 144,
"split": "train"
} |
[
{
"content": "Question:\nThe points $(0,0)\\,$ , $(a,11)\\,$ , and $(b,37)\\,$ are the vertices of an equilateral triangle. Find the value of $ab\\,$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 315 | {
"ability": "math",
"data_source": "aime",
"index": 145,
"split": "train"
} |
[
{
"content": "Question:\nA solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the play... | 394 | {
"ability": "math",
"data_source": "aime",
"index": 146,
"split": "train"
} |
[
{
"content": "Question:\nIn triangle $ABC,\\,$ angle $C$ is a right angle and the altitude from $C\\,$ meets $\\overline{AB}\\,$ at $D.\\,$ The lengths of the sides of $\\triangle ABC\\,$ are integers, $BD=29^3,\\,$ and $\\cos B=m/n\\,$ , where $m\\,$ and $n\\,$ are relatively prime positive integers. Find $m+... | 450 | {
"ability": "math",
"data_source": "aime",
"index": 147,
"split": "train"
} |
[
{
"content": "Question:\nNinety-four bricks, each measuring $4''\\times10''\\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\\,$ or $10''\\,$ or $19''\\,$ to the total height of the tower. How many different tower heights ca... | 465 | {
"ability": "math",
"data_source": "aime",
"index": 148,
"split": "train"
} |
[
{
"content": "Question:\nA fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be par... | 702 | {
"ability": "math",
"data_source": "aime",
"index": 149,
"split": "train"
} |
[
{
"content": "Question:\nA beam of light strikes $\\overline{BC}\\,$ at point $C\\,$ with angle of incidence $\\alpha=19.94^\\circ\\,$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\\overline{AB}\\,$ and $\\overline{BC}\\,$ according t... | 71 | {
"ability": "math",
"data_source": "aime",
"index": 150,
"split": "train"
} |
[
{
"content": "Question:\nGiven a point $P^{}_{}$ on a triangular piece of paper $ABC,\\,$ consider the creases that are formed in the paper when $A, B,\\,$ and $C\\,$ are folded onto $P.\\,$ Let us call $P_{}^{}$ a fold point of $\\triangle ABC\\,$ if these creases, which number three unless $P^{}_{}$ is one of... | 597 | {
"ability": "math",
"data_source": "aime",
"index": 151,
"split": "train"
} |
[
{
"content": "Question:\nSquare $S_{1}$ is $1\\times 1.$ For $i\\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square ... | 255 | {
"ability": "math",
"data_source": "aime",
"index": 152,
"split": "train"
} |
[
{
"content": "Question:\nFind the last three digits of the product of the positive roots of $\\sqrt{1995}x^{\\log_{1995}x}=x^2.$\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 25 | {
"ability": "math",
"data_source": "aime",
"index": 153,
"split": "train"
} |
[
{
"content": "Question:\nStarting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in... | 67 | {
"ability": "math",
"data_source": "aime",
"index": 154,
"split": "train"
} |
[
{
"content": "Question:\nCircles of radius $3$ and $6$ are externally tangent to each other and are internally tangent to a circle of radius $9$ . The circle of radius $9$ has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.\nPlease reason step by ... | 224 | {
"ability": "math",
"data_source": "aime",
"index": 155,
"split": "train"
} |
[
{
"content": "Question:\nFor certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i=\\sqrt{-1}.$ Find $b.$\nPlease reason step by step, and put your final answe... | 51 | {
"ability": "math",
"data_source": "aime",
"index": 156,
"split": "train"
} |
[
{
"content": "Question:\nLet $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n_{}$ but do not divide $n_{}$ ?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 589 | {
"ability": "math",
"data_source": "aime",
"index": 157,
"split": "train"
} |
[
{
"content": "Question:\nFor how many ordered pairs of positive integers $(x,y),$ with $y<x\\le 100,$ are both $\\frac xy$ and $\\frac{x+1}{y+1}$ integers?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 85 | {
"ability": "math",
"data_source": "aime",
"index": 158,
"split": "train"
} |
[
{
"content": "Question:\nTriangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\\overline{AM}$ with $AD=10$ and $\\angle BDC=3\\angle BAC.$ Then the perimeter of $\\triangle ABC$ may be written in the form $a+\\sqrt{b},$ where $a$ and $b$ are integers. Find $a+b.... | 616 | {
"ability": "math",
"data_source": "aime",
"index": 159,
"split": "train"
} |
[
{
"content": "Question:\nWhat is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 215 | {
"ability": "math",
"data_source": "aime",
"index": 160,
"split": "train"
} |
[
{
"content": "Question:\nA right rectangular prism $P_{}$ (i.e., a rectangular parallelepiped) has sides of integral length $a, b, c,$ with $a\\le b\\le c.$ A plane parallel to one of the faces of $P_{}$ cuts $P_{}$ into two prisms, one of which is similar to $P_{},$ and both of which have nonzero volume. Give... | 40 | {
"ability": "math",
"data_source": "aime",
"index": 161,
"split": "train"
} |
[
{
"content": "Question:\nPyramid $OABCD$ has square base $ABCD,$ congruent edges $\\overline{OA}, \\overline{OB}, \\overline{OC},$ and $\\overline{OD},$ and $\\angle AOB=45^\\circ.$ Let $\\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\\cos \\theta=m+\\sqrt{n},$ where ... | 5 | {
"ability": "math",
"data_source": "aime",
"index": 162,
"split": "train"
} |
[
{
"content": "Question:\nLet $f(n)$ be the integer closest to $\\sqrt[4]{n}.$ Find $\\sum_{k=1}^{1995}\\frac 1{f(k)}.$\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 400 | {
"ability": "math",
"data_source": "aime",
"index": 163,
"split": "train"
} |
[
{
"content": "Question:\nIn 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 expr... | 378 | {
"ability": "math",
"data_source": "aime",
"index": 164,
"split": "train"
} |
[
{
"content": "Question:\nLet $p_{}$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p_{}$ can be written in the form $m/n$ where $m_{}$ and $n_{}$ are relatively prime positive integers, find $m+... | 37 | {
"ability": "math",
"data_source": "aime",
"index": 165,
"split": "train"
} |
[
{
"content": "Question:\nFor each real number $x$ , let $\\lfloor x \\rfloor$ denote the greatest integer that does not exceed $x$ . For how many positive integers $n$ is it true that $n<1000$ and that $\\lfloor \\log_{2} n \\rfloor$ is a positive even integer?\nPlease reason step by step, and put your final an... | 340 | {
"ability": "math",
"data_source": "aime",
"index": 166,
"split": "train"
} |
[
{
"content": "Question:\nFind the smallest positive integer $n$ for which the expansion of $(xy-3x+7y-21)^n$ , after like terms have been collected, has at least 1996 terms.\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 44 | {
"ability": "math",
"data_source": "aime",
"index": 167,
"split": "train"
} |
[
{
"content": "Question:\nA wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of a shadow, which does not include the area bene... | 166 | {
"ability": "math",
"data_source": "aime",
"index": 168,
"split": "train"
} |
[
{
"content": "Question:\nSuppose that the roots of $x^3+3x^2+4x-11=0$ are $a$ , $b$ , and $c$ , and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$ , $b+c$ , and $c+a$ . Find $t$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 23 | {
"ability": "math",
"data_source": "aime",
"index": 169,
"split": "train"
} |
[
{
"content": "Question:\nIn a five-team tournament, each team plays one game with every other team. Each team has a $50\\%$ chance of winning any game it plays. (There are no ties.) Let $\\dfrac{m}{n}$ be the probability that the tournament will produce neither an undefeated team nor a winless team, where $m$ a... | 49 | {
"ability": "math",
"data_source": "aime",
"index": 170,
"split": "train"
} |
[
{
"content": "Question:\nTwo squares of a $7\\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane board. How many inequivalent color schemes are possible?\nPlease reason step by step,... | 300 | {
"ability": "math",
"data_source": "aime",
"index": 171,
"split": "train"
} |
[
{
"content": "Question:\nThe harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$ ?\nPlease reason step by step, and put your final answer wit... | 799 | {
"ability": "math",
"data_source": "aime",
"index": 172,
"split": "train"
} |
[
{
"content": "Question:\nA bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens t... | 342 | {
"ability": "math",
"data_source": "aime",
"index": 173,
"split": "train"
} |
[
{
"content": "Question:\nFind the smallest positive integer solution to $\\tan{19x^{\\circ}}=\\dfrac{\\cos{96^{\\circ}}+\\sin{96^{\\circ}}}{\\cos{96^{\\circ}}-\\sin{96^{\\circ}}}$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 159 | {
"ability": "math",
"data_source": "aime",
"index": 174,
"split": "train"
} |
[
{
"content": "Question:\nLet $\\mathrm {P}$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have a positive imaginary part, and suppose that $\\mathrm {P}=r(\\cos{\\theta^{\\circ}}+i\\sin{\\theta^{\\circ}})$ , where $0<r$ and $0\\leq \\theta <360$ . Find $\\theta$ .\nPlease reason step by step, and pu... | 276 | {
"ability": "math",
"data_source": "aime",
"index": 175,
"split": "train"
} |
[
{
"content": "Question:\nFor each permutation $a_1,a_2,a_3,\\cdots,a_{10}$ of the integers $1,2,3,\\cdots,10$ , form the sum $|a_1-a_2|+|a_3-a_4|+|a_5-a_6|+|a_7-a_8|+|a_9-a_{10}|$ . The average value of all such sums can be written in the form $\\dfrac{p}{q}$ , where $p$ and $q$ are relatively prime positive in... | 58 | {
"ability": "math",
"data_source": "aime",
"index": 176,
"split": "train"
} |
[
{
"content": "Question:\nIn triangle $ABC$ , $AB=\\sqrt{30}$ , $AC=\\sqrt{6}$ , and $BC=\\sqrt{15}$ . There is a point $D$ for which $\\overline{AD}$ bisects $\\overline{BC}$ , and $\\angle ADB$ is a right angle. The ratio \\[\\dfrac{\\text{Area}(\\triangle ADB)}{\\text{Area}(\\triangle ABC)}\\] can be written ... | 65 | {
"ability": "math",
"data_source": "aime",
"index": 177,
"split": "train"
} |
[
{
"content": "Question:\nA $150\\times 324\\times 375$ rectangular solid is made by gluing together $1\\times 1\\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\\times 1\\times 1$ cubes?\nPlease reason step by step, and put your final answer within \\boxed{ ... | 768 | {
"ability": "math",
"data_source": "aime",
"index": 178,
"split": "train"
} |
[
{
"content": "Question:\nIn parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\\overline{AC}$ and $\\overline{BD}$ . Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB$ . Find the greatest integer that does not exceed $1000r$ .\nPleas... | 777 | {
"ability": "math",
"data_source": "aime",
"index": 179,
"split": "train"
} |
[
{
"content": "Question:\nHow many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 750 | {
"ability": "math",
"data_source": "aime",
"index": 180,
"split": "train"
} |
[
{
"content": "Question:\nThe nine horizontal and nine vertical lines on an $8\\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$\nPlease reason step by step, and put you... | 125 | {
"ability": "math",
"data_source": "aime",
"index": 181,
"split": "train"
} |
[
{
"content": "Question:\nSarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should... | 126 | {
"ability": "math",
"data_source": "aime",
"index": 182,
"split": "train"
} |
[
{
"content": "Question:\nCircles of radii 5, 5, 8, and $m/n$ are mutually externally tangent, 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": "user"
}
] | 17 | {
"ability": "math",
"data_source": "aime",
"index": 183,
"split": "train"
} |
[
{
"content": "Question:\nThe number $r$ can be expressed as a four-place decimal $0.abcd,$ where $a, b, c,$ and $d$ represent digits, any of which could be zero. It is desired to approximate $r$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to $r$ is $\\... | 417 | {
"ability": "math",
"data_source": "aime",
"index": 184,
"split": "train"
} |
[
{
"content": "Question:\nPoint $B$ is in the exterior of the regular $n$ -sided polygon $A_1A_2\\cdots A_n$ , and $A_1A_2B$ is an equilateral triangle. What is the largest value of $n$ for which $A_1$ , $A_n$ , and $B$ are consecutive vertices of a regular polygon?\nPlease reason step by step, and put your fina... | 42 | {
"ability": "math",
"data_source": "aime",
"index": 185,
"split": "train"
} |
[
{
"content": "Question:\nA car travels due east at $\\frac 23$ miles per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\\frac 12\\sqrt{2}$ miles per minute. At time $t=0$ , the center of the storm is $110$ miles due north of the car. At time... | 198 | {
"ability": "math",
"data_source": "aime",
"index": 186,
"split": "train"
} |
[
{
"content": "Question:\nHow many different $4\\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 90 | {
"ability": "math",
"data_source": "aime",
"index": 187,
"split": "train"
} |
[
{
"content": "Question:\nGiven a nonnegative real number $x$ , let $\\langle x\\rangle$ denote the fractional part of $x$ ; that is, $\\langle x\\rangle=x-\\lfloor x\\rfloor$ , where $\\lfloor x\\rfloor$ denotes the greatest integer less than or equal to $x$ . Suppose that $a$ is positive, $\\langle a^{-1}\\ran... | 233 | {
"ability": "math",
"data_source": "aime",
"index": 188,
"split": "train"
} |
[
{
"content": "Question:\nEvery card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combi... | 117 | {
"ability": "math",
"data_source": "aime",
"index": 189,
"split": "train"
} |
[
{
"content": "Question:\nLet $x=\\frac{\\sum\\limits_{n=1}^{44} \\cos n^\\circ}{\\sum\\limits_{n=1}^{44} \\sin n^\\circ}$ . What is the greatest integer that does not exceed $100x$ ?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 241 | {
"ability": "math",
"data_source": "aime",
"index": 190,
"split": "train"
} |
[
{
"content": "Question:\nThe function $f$ defined by $f(x)= \\frac{ax+b}{cx+d}$ , where $a$ , $b$ , $c$ and $d$ are nonzero real numbers, has the properties $f(19)=19$ , $f(97)=97$ and $f(f(x))=x$ for all values except $\\frac{-d}{c}$ . Find the unique number that is not in the range of $f$ .\nPlease reason ste... | 58 | {
"ability": "math",
"data_source": "aime",
"index": 191,
"split": "train"
} |
[
{
"content": "Question:\nLet $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$ . Let $m/n$ be the probability that $\\sqrt{2+\\sqrt{3}}\\le |v+w|$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .\nPlease reason step by step, and put your final answer within \\... | 582 | {
"ability": "math",
"data_source": "aime",
"index": 192,
"split": "train"
} |
[
{
"content": "Question:\nThe sides of rectangle $ABCD$ have lengths $10$ and $11$ . An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$ . The maximum possible area of such a triangle can be written in the form $p\\sqrt{q}-r$ , where $p$ , $q$ , and $r$ are positive integers, an... | 554 | {
"ability": "math",
"data_source": "aime",
"index": 193,
"split": "train"
} |
[
{
"content": "Question:\nFor how many values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^6$ and $8^8$ , and $k$ ?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 25 | {
"ability": "math",
"data_source": "aime",
"index": 194,
"split": "train"
} |
[
{
"content": "Question:\nFind the number of ordered pairs $(x,y)$ of positive integers that satisfy $x \\le 2y \\le 60$ and $y \\le 2x \\le 60$ .\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 480 | {
"ability": "math",
"data_source": "aime",
"index": 195,
"split": "train"
} |
[
{
"content": "Question:\nThe graph of $y^2 + 2xy + 40|x|= 400$ partitions the plane into several regions. What is the area of the bounded region?\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 800 | {
"ability": "math",
"data_source": "aime",
"index": 196,
"split": "train"
} |
[
{
"content": "Question:\nNine tiles are numbered $1, 2, 3, \\cdots, 9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers . ... | 17 | {
"ability": "math",
"data_source": "aime",
"index": 197,
"split": "train"
} |
[
{
"content": "Question:\nGiven that $A_k = \\frac {k(k - 1)}2\\cos\\frac {k(k - 1)\\pi}2,$ find $|A_{19} + A_{20} + \\cdots + A_{98}|.$\nPlease reason step by step, and put your final answer within \\boxed{ }.",
"role": "user"
}
] | 40 | {
"ability": "math",
"data_source": "aime",
"index": 198,
"split": "train"
} |
[
{
"content": "Question:\nLet $ABCD$ be a parallelogram . Extend $\\overline{DA}$ through $A$ to a point $P,$ and let $\\overline{PC}$ meet $\\overline{AB}$ at $Q$ and $\\overline{DB}$ at $R.$ Given that $PQ = 735$ and $QR = 112,$ find $RC.$\nPlease reason step by step, and put your final answer within \\boxed{... | 308 | {
"ability": "math",
"data_source": "aime",
"index": 199,
"split": "train"
} |
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