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275 | For polynomial $P(x)=1-\dfrac{1}{3}x+\dfrac{1}{6}x^{2}$ , define $Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\sum_{i=0}^{50} a_ix^{i}$ .
Then $\sum_{i=0}^{50} |a_i|=\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2016-II-6', 'Year': 2016, 'Problem Number': 6, 'Part': 'II'} | null | null | null |
840 | Squares $ABCD$ 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 a side of $IJKL$ . Find the difference between the largest and smallest positive integer values for the area of $IJKL$ . | math | qq8933/AIME_1983_2024 | {'ID': '2016-II-7', 'Year': 2016, 'Problem Number': 7, 'Part': 'II'} | null | null | null |
728 | Find 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 $61$ . | math | qq8933/AIME_1983_2024 | {'ID': '2016-II-8', 'Year': 2016, 'Problem Number': 8, 'Part': 'II'} | null | null | null |
262 | The 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 $c_k$ . | math | qq8933/AIME_1983_2024 | {'ID': '2016-II-9', 'Year': 2016, 'Problem Number': 9, 'Part': 'II'} | null | null | null |
43 | Triangle $ABC$ 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 relatively prime positive integers. Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2016-II-10', 'Year': 2016, 'Problem Number': 10, 'Part': 'II'} | null | null | null |
749 | For 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. | math | qq8933/AIME_1983_2024 | {'ID': '2016-II-11', 'Year': 2016, 'Problem Number': 11, 'Part': 'II'} | null | null | null |
732 | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color. [asy] draw(Circle((0,0), 4)); draw(Circle((0,0), 3)); draw((0,4)--(0,3)); draw((0,-4)--(0,-3)); draw((-2.598, 1.5)--(-3.4641, 2)); draw((-2.598, -1.5)--(-3.4641, -2)); draw((2.598, -1.5)--(3.4641, -2)); draw((2.598, 1.5)--(3.4641, 2)); [/asy] | math | qq8933/AIME_1983_2024 | {'ID': '2016-II-12', 'Year': 2016, 'Problem Number': 12, 'Part': 'II'} | null | null | null |
371 | Beatrix 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 an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ . | math | qq8933/AIME_1983_2024 | {'ID': '2016-II-13', 'Year': 2016, 'Problem Number': 13, 'Part': 'II'} | null | null | null |
450 | Equilateral $\triangle ABC$ 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 angle (the angle between the two planes). There is a point $O$ whose distance from each of $A,B,C,P,$ and $Q$ is $d$ . Find $d$ . | math | qq8933/AIME_1983_2024 | {'ID': '2016-II-14', 'Year': 2016, 'Problem Number': 14, 'Part': 'II'} | null | null | null |
863 | For $1 \leq i \leq 215$ 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 $\sum_{1 \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}$ . The maximum possible value of $x_2=\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2016-II-15', 'Year': 2016, 'Problem Number': 15, 'Part': 'II'} | null | null | null |
390 | Fifteen 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 vertices are among these $15$ points. | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-1', 'Year': 2017, 'Problem Number': 1, 'Part': 'I'} | null | null | null |
62 | When 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 $m+n+r+s$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-2', 'Year': 2017, 'Problem Number': 2, 'Part': 'I'} | null | null | null |
69 | For 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 $1000$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-3', 'Year': 2017, 'Problem Number': 3, 'Part': 'I'} | null | null | null |
803 | A 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, and $n$ is not divisible by the square of any prime. Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-4', 'Year': 2017, 'Problem Number': 4, 'Part': 'I'} | null | null | null |
321 | A 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 $\underline{a} \underline{b} \underline{c}$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-5', 'Year': 2017, 'Problem Number': 5, 'Part': 'I'} | null | null | null |
48 | A 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 $\frac{14}{25}$ . Find the difference between the largest and smallest possible values of $x$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-6', 'Year': 2017, 'Problem Number': 6, 'Part': 'I'} | null | null | null |
564 | For 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 $1000$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-7', 'Year': 2017, 'Problem Number': 7, 'Part': 'I'} | null | null | null |
41 | Two 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 $a$ and $b$ respectively, and $\angle OQP$ and $\angle ORP$ are both right angles. The probability that $QR \leq 100$ is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-8', 'Year': 2017, 'Problem Number': 8, 'Part': 'I'} | null | null | null |
45 | Let $a_{10} = 10$ , 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 $99$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-9', 'Year': 2017, 'Problem Number': 9, 'Part': 'I'} | null | null | null |
56 | Let $z_1 = 18 + 83i$ , $z_2 = 18 + 39i,$ 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. Find the real part of $z$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-10', 'Year': 2017, 'Problem Number': 10, 'Part': 'I'} | null | null | null |
360 | Consider 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 number of arrangements for which $m = 5$ . Find the remainder when $Q$ is divided by $1000$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-11', 'Year': 2017, 'Problem Number': 11, 'Part': 'I'} | null | null | null |
252 | Call 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 product-free subsets of the set $\{1, 2, 3, 4, \ldots, 7, 8, 9, 10\}$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-12', 'Year': 2017, 'Problem Number': 12, 'Part': 'I'} | null | null | null |
59 | For 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 $1000$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-13', 'Year': 2017, 'Problem Number': 13, 'Part': 'I'} | null | null | null |
896 | Let $a > 1$ 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 $1000$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-14', 'Year': 2017, 'Problem Number': 14, 'Part': 'I'} | null | null | null |
145 | The 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 $p$ is not divisible by the square of any prime. Find $m+n+p$ . [asy] size(5cm); pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); real t = .385, s = 3.5*t-1; pair R = A*t+B*(1-t), P=B*s; pair Q = dir(-60) * (R-P) + P; fill(P--Q--R--cycle,gray); draw(A--B--C--A^^P--Q--R--P); dot(A--B--C--P--Q--R); [/asy] | math | qq8933/AIME_1983_2024 | {'ID': '2017-I-15', 'Year': 2017, 'Problem Number': 15, 'Part': 'I'} | null | null | null |
196 | Find the number of subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ that are subsets of neither $\{1, 2, 3, 4, 5\}$ nor $\{4, 5, 6, 7, 8\}$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-1', 'Year': 2017, 'Problem Number': 1, 'Part': 'II'} | null | null | null |
781 | Teams $T_1$ , $T_2$ , $T_3$ , 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 is $\frac{i}{i+j}$ , and the outcomes of all the matches are independent. The probability that $T_4$ will be the champion is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-2', 'Year': 2017, 'Problem Number': 2, 'Part': 'II'} | null | null | null |
409 | A 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. Find $p+q$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-3', 'Year': 2017, 'Problem Number': 3, 'Part': 'II'} | null | null | null |
222 | Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-4', 'Year': 2017, 'Problem Number': 4, 'Part': 'II'} | null | null | null |
791 | A 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 $x+y$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-5', 'Year': 2017, 'Problem Number': 5, 'Part': 'II'} | null | null | null |
195 | Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer. | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-6', 'Year': 2017, 'Problem Number': 6, 'Part': 'II'} | null | null | null |
501 | Find 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. | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-7', 'Year': 2017, 'Problem Number': 7, 'Part': 'II'} | null | null | null |
134 | Find 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. | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-8', 'Year': 2017, 'Problem Number': 8, 'Part': 'II'} | null | null | null |
13 | A 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 of each color and at least one card with each number, the probability that Sharon can discard one of her cards and $\textit{still}$ have at least one card of each color and at least one card with each number is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-9', 'Year': 2017, 'Problem Number': 9, 'Part': 'II'} | null | null | null |
546 | Rectangle $ABCD$ 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 $BCON$ , and $\overline{BP}$ bisects the area of $BCON$ . Find the area of $\triangle CDP$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-10', 'Year': 2017, 'Problem Number': 10, 'Part': 'II'} | null | null | null |
544 | Five 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 through other towns on the way). | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-11', 'Year': 2017, 'Problem Number': 11, 'Part': 'II'} | null | null | null |
110 | Circle $C_0$ 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 radius $r^2$ and is internally tangent to $C_1$ at point $A_1$ . In this way a sequence of circles $C_1,C_2,C_3,\ldots$ and a sequence of points on the circles $A_1,A_2,A_3,\ldots$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$ , and point $A_n$ lies on $C_n$ $90^{\circ}$ counterclockwise from point $A_{n-1}$ , as shown in the figure below. There is one point $B$ inside all of these circles. When $r = \frac{11}{60}$ , the distance from the center $C_0$ to $B$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . [asy] draw(Circle((0,0),125)); draw(Circle((25,0),100)); draw(Circle((25,20),80)); draw(Circle((9,20),64)); dot((125,0)); label("$A_0$",(125,0),E); dot((25,100)); label("$A_1$",(25,100),SE); dot((-55,20)); label("$A_2$",(-55,20),E); [/asy] | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-12', 'Year': 2017, 'Problem Number': 12, 'Part': 'II'} | null | null | null |
245 | For 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 $f(n+1)=f(n)+78$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-13', 'Year': 2017, 'Problem Number': 13, 'Part': 'II'} | null | null | null |
168 | A $10\times10\times10$ 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. | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-14', 'Year': 2017, 'Problem Number': 14, 'Part': 'II'} | null | null | null |
682 | Tetrahedron $ABCD$ 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. Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2017-II-15', 'Year': 2017, 'Problem Number': 15, 'Part': 'II'} | null | null | null |
600 | Let $S$ 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 by $1000$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-1', 'Year': 2018, 'Problem Number': 1, 'Part': 'I'} | null | null | null |
925 | The 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 $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$ , where $a > 0$ . Find the base- $10$ representation of $n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-2', 'Year': 2018, 'Problem Number': 2, 'Part': 'I'} | null | null | null |
157 | Kathy 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, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-3', 'Year': 2018, 'Problem Number': 3, 'Part': 'I'} | null | null | null |
289 | In $\triangle ABC, AB = AC = 10$ 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 relatively prime positive integers. Find $p+q$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-4', 'Year': 2018, 'Problem Number': 4, 'Part': 'I'} | null | null | null |
189 | For 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 $K$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-5', 'Year': 2018, 'Problem Number': 5, 'Part': 'I'} | null | null | null |
440 | Let $N$ 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 $1000$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-6', 'Year': 2018, 'Problem Number': 6, 'Part': 'I'} | null | null | null |
52 | A 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). | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-7', 'Year': 2018, 'Problem Number': 7, 'Part': 'I'} | null | null | null |
147 | Let $ABCDEF$ 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 $d^2$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-8', 'Year': 2018, 'Problem Number': 8, 'Part': 'I'} | null | null | null |
210 | Find 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. | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-9', 'Year': 2018, 'Problem Number': 9, 'Part': 'I'} | null | null | null |
4 | The 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 circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path $AJABCHCHIJA$ , which has $10$ steps. Let $n$ be the number of paths with $15$ steps that begin and end at point $A.$ Find the remainder when $n$ is divided by $1000$ . [asy] size(6cm); draw(unitcircle); draw(scale(2) * unitcircle); for(int d = 90; d < 360 + 90; d += 72){ draw(2 * dir(d) -- dir(d)); } dot(1 * dir( 90), linewidth(5)); dot(1 * dir(162), linewidth(5)); dot(1 * dir(234), linewidth(5)); dot(1 * dir(306), linewidth(5)); dot(1 * dir(378), linewidth(5)); dot(2 * dir(378), linewidth(5)); dot(2 * dir(306), linewidth(5)); dot(2 * dir(234), linewidth(5)); dot(2 * dir(162), linewidth(5)); dot(2 * dir( 90), linewidth(5)); label("$A$", 1 * dir( 90), -dir( 90)); label("$B$", 1 * dir(162), -dir(162)); label("$C$", 1 * dir(234), -dir(234)); label("$D$", 1 * dir(306), -dir(306)); label("$E$", 1 * dir(378), -dir(378)); label("$F$", 2 * dir(378), dir(378)); label("$G$", 2 * dir(306), dir(306)); label("$H$", 2 * dir(234), dir(234)); label("$I$", 2 * dir(162), dir(162)); label("$J$", 2 * dir( 90), dir( 90)); [/asy] | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-10', 'Year': 2018, 'Problem Number': 10, 'Part': 'I'} | null | null | null |
195 | Find the least positive integer $n$ such that when $3^n$ is written in base $143$ , its two right-most digits in base $143$ are $01$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-11', 'Year': 2018, 'Problem Number': 11, 'Part': 'I'} | null | null | null |
683 | For 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 $n$ are relatively prime positive integers. Find $m$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-12', 'Year': 2018, 'Problem Number': 12, 'Part': 'I'} | null | null | null |
126 | Let $\triangle ABC$ 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 $X$ varies along $\overline{BC}$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-13', 'Year': 2018, 'Problem Number': 13, 'Part': 'I'} | null | null | null |
351 | Let $SP_1P_2P_3EP_4P_5$ 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 of no more than $12$ jumps that end at $E$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-14', 'Year': 2018, 'Problem Number': 14, 'Part': 'I'} | null | null | null |
59 | David 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 quadrilateral $A$ , and define $\varphi_B$ and $\varphi_C$ similarly. Suppose that $\sin\varphi_A=\frac{2}{3}$ , $\sin\varphi_B=\frac{3}{5}$ , and $\sin\varphi_C=\frac{6}{7}$ . All three quadrilaterals have the same area $K$ , which can be written in the form $\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-I-15', 'Year': 2018, 'Problem Number': 15, 'Part': 'I'} | null | null | null |
800 | Points $A$ , $B$ , 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 $C$ , Paul starting at $B$ and running toward $C$ , and Eve starting at $C$ and running toward $A$ . When Paul meets Eve, he turns around and runs toward $A$ . Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-1', 'Year': 2018, 'Problem Number': 1, 'Part': 'II'} | null | null | null |
112 | Let $a_{0} = 2$ , $a_{1} = 5$ , 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 $a_{2018} \cdot a_{2020} \cdot a_{2022}$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-2', 'Year': 2018, 'Problem Number': 2, 'Part': 'II'} | null | null | null |
371 | Find 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. | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-3', 'Year': 2018, 'Problem Number': 3, 'Part': 'II'} | null | null | null |
23 | In 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 regions. Then $K = \frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-4', 'Year': 2018, 'Problem Number': 4, 'Part': 'II'} | null | null | null |
74 | Suppose 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 $a^2 + b^2$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-5', 'Year': 2018, 'Problem Number': 5, 'Part': 'II'} | null | null | null |
37 | A 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 integers. Find $m + n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-6', 'Year': 2018, 'Problem Number': 6, 'Part': 'II'} | null | null | null |
20 | Triangle $ABC$ 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 $A = Q_{0}, Q_{1}, Q_{2}, ... , Q_{2450} = C$ are on segment $\overline{AC}$ with $Q_{k}$ between $Q_{k-1}$ and $Q_{k+1}$ for $k = 1, 2, ..., 2449$ . Furthermore, each segment $\overline{P_{k}Q_{k}}$ , $k = 1, 2, ..., 2449$ , is parallel to $\overline{BC}$ . The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions has the same area. Find the number of segments $\overline{P_{k}Q_{k}}$ , $k = 1, 2, ..., 2450$ , that have rational length. | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-7', 'Year': 2018, 'Problem Number': 7, 'Part': 'II'} | null | null | null |
556 | A 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 $(4, 4)$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-8', 'Year': 2018, 'Problem Number': 8, 'Part': 'II'} | null | null | null |
184 | Octagon $ABCDEFGH$ 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 $\overline{AH}$ , and partition the octagon into 7 triangles by drawing segments $\overline{JB}$ , $\overline{JC}$ , $\overline{JD}$ , $\overline{JE}$ , $\overline{JF}$ , and $\overline{JG}$ . Find the area of the convex polygon whose vertices are the centroids of these 7 triangles. [asy] unitsize(6); pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0); pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17); draw(P--Q--R--SS--cycle); draw(J--B); draw(J--C); draw(J--D); draw(J--EE); draw(J--F); draw(J--G); draw(A--B); draw(H--G); real dark = 0.6; filldraw(A--B--P--cycle, gray(dark)); filldraw(H--G--Q--cycle, gray(dark)); filldraw(F--EE--R--cycle, gray(dark)); filldraw(D--C--SS--cycle, gray(dark)); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(J); dot(H); defaultpen(fontsize(10pt)); real r = 1.3; label("$A$", A, W*r); label("$B$", B, S*r); label("$C$", C, S*r); label("$D$", D, E*r); label("$E$", EE, E*r); label("$F$", F, N*r); label("$G$", G, N*r); label("$H$", H, W*r); label("$J$", J, W*r); [/asy] | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-9', 'Year': 2018, 'Problem Number': 9, 'Part': 'II'} | null | null | null |
756 | Find 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 $\{1, 2, 3, 4, 5\}$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-10', 'Year': 2018, 'Problem Number': 10, 'Part': 'II'} | null | null | null |
461 | Find 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 $k$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-11', 'Year': 2018, 'Problem Number': 11, 'Part': 'II'} | null | null | null |
112 | Let $ABCD$ 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 area of quadrilateral $ABCD$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-12', 'Year': 2018, 'Problem Number': 12, 'Part': 'II'} | null | null | null |
647 | Misha 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 $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-13', 'Year': 2018, 'Problem Number': 13, 'Part': 'II'} | null | null | null |
227 | The 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 $\omega$ at $Y$ . Assume that $AP = 3$ , $PB = 4$ , $AC = 8$ , and $AQ = \dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-14', 'Year': 2018, 'Problem Number': 14, 'Part': 'II'} | null | null | null |
185 | Find 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 $\{0, 1, 2, 3, 4, 5, 6\}$ . | math | qq8933/AIME_1983_2024 | {'ID': '2018-II-15', 'Year': 2018, 'Problem Number': 15, 'Part': 'II'} | null | null | null |
342 | Consider the integer \[N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.\] Find the sum of the digits of $N$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-1', 'Year': 2019, 'Problem Number': 1, 'Part': 'I'} | null | null | null |
29 | Jenn 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 $n$ are relatively prime positive integers. Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-2', 'Year': 2019, 'Problem Number': 2, 'Part': 'I'} | null | null | null |
120 | In $\triangle PQR$ , $PR=15$ , $QR=20$ , and $PQ=25$ . Points $A$ and $B$ lie on $\overline{PQ}$ , points $C$ and $D$ lie on $\overline{QR}$ , and points $E$ and $F$ lie on $\overline{PR}$ , with $PA=QB=QC=RD=RE=PF=5$ . Find the area of hexagon $ABCDEF$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-3', 'Year': 2019, 'Problem Number': 3, 'Part': 'I'} | null | null | null |
122 | A 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. No player removed from the game may reenter the game, although a substitute entering the game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let $n$ be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Find the remainder when $n$ is divided by $1000$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-4', 'Year': 2019, 'Problem Number': 4, 'Part': 'I'} | null | null | null |
252 | A 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 $\tfrac{1}{3}$ , independently of its previous moves. The probability that it will hit the coordinate axes at $(0,0)$ is $\tfrac{m}{3^n}$ , where $m$ and $n$ are positive integers, and $m$ is not divisible by $3$ . Find $m + n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-5', 'Year': 2019, 'Problem Number': 5, 'Part': 'I'} | null | null | null |
90 | In 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 $\overline{KM}$ at $O$ with $KO = 8$ . Find $MO$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-6', 'Year': 2019, 'Problem Number': 6, 'Part': 'I'} | null | null | null |
880 | There 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 $x$ , and let $n$ be the number of (not necessarily distinct) prime factors in the prime factorization of $y$ . Find $3m+2n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-7', 'Year': 2019, 'Problem Number': 7, 'Part': 'I'} | null | null | null |
67 | Let $x$ 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 $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-8', 'Year': 2019, 'Problem Number': 8, 'Part': 'I'} | null | null | null |
540 | Let $\tau(n)$ denote the number of positive integer divisors of $n$ . Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-9', 'Year': 2019, 'Problem Number': 9, 'Part': 'I'} | null | null | null |
352 | For 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 \[\left| \sum_{1 \le j <k \le 673} z_jz_k \right|\] can be expressed in the form $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-10', 'Year': 2019, 'Problem Number': 10, 'Part': 'I'} | null | null | null |
20 | In $\triangle ABC$ , 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 extensions of the other two sides. Suppose that the excircle tangent to $\overline{BC}$ is internally tangent to $\omega$ , and the other two excircles are both externally tangent to $\omega$ . Find the minimum possible value of the perimeter of $\triangle ABC$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-11', 'Year': 2019, 'Problem Number': 11, 'Part': 'I'} | null | null | null |
230 | Given $f(z) = z^2-19z$ , 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 $m+\sqrt{n}+11i$ . Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-12', 'Year': 2019, 'Problem Number': 12, 'Part': 'I'} | null | null | null |
32 | Triangle $ABC$ 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 $\tfrac{a+b\sqrt{c}}{d}$ , where $a$ , $b$ , $c$ , and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-13', 'Year': 2019, 'Problem Number': 13, 'Part': 'I'} | null | null | null |
97 | Find the least odd prime factor of $2019^8 + 1$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-14', 'Year': 2019, 'Problem Number': 14, 'Part': 'I'} | null | null | null |
65 | Let $\overline{AB}$ 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 $\omega$ . Circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$ . Line $PQ$ intersects $\omega$ at $X$ and $Y$ . Assume that $AP=5$ , $PB=3$ , $XY=11$ , and $PQ^2 = \tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-I-15', 'Year': 2019, 'Problem Number': 15, 'Part': 'I'} | null | null | null |
59 | Two 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 positive integers. Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-1', 'Year': 2019, 'Problem Number': 1, 'Part': 'II'} | null | null | null |
107 | Lily 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 visits pad $7$ is $\tfrac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-2', 'Year': 2019, 'Problem Number': 2, 'Part': 'II'} | null | null | null |
96 | Find the number of $7$ -tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following system of equations: \[abc=70\] \[cde=71\] \[efg=72.\] | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-3', 'Year': 2019, 'Problem Number': 3, 'Part': 'II'} | null | null | null |
187 | A 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 $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-4', 'Year': 2019, 'Problem Number': 4, 'Part': 'II'} | null | null | null |
520 | Four 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 $8$ people to be seated at the table under these conditions. Find the remainder when $N$ is divided by $1000$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-5', 'Year': 2019, 'Problem Number': 5, 'Part': 'II'} | null | null | null |
216 | In 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 solution $x>1$ . Find $b$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-6', 'Year': 2019, 'Problem Number': 6, 'Part': 'II'} | null | null | null |
715 | Triangle $ABC$ 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 $\triangle ABC$ are segments of lengths $55,45$ , and $15$ , respectively. Find the perimeter of the triangle whose sides lie on lines $\ell_A,\ell_B$ , and $\ell_C$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-7', 'Year': 2019, 'Problem Number': 7, 'Part': 'II'} | null | null | null |
53 | The 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 $1000$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-8', 'Year': 2019, 'Problem Number': 8, 'Part': 'II'} | null | null | null |
472 | Call a positive integer $n$ $k$ - pretty if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$ . For example, $18$ is $6$ -pretty. Let $S$ be the sum of the positive integers less than $2019$ that are $20$ -pretty. Find $\tfrac{S}{20}$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-9', 'Year': 2019, 'Problem Number': 9, 'Part': 'II'} | null | null | null |
547 | There 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 relatively prime positive integers. Find $p+q$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-10', 'Year': 2019, 'Problem Number': 10, 'Part': 'II'} | null | null | null |
11 | Triangle $ABC$ 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 $A.$ Then $AK=\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-11', 'Year': 2019, 'Problem Number': 11, 'Part': 'II'} | null | null | null |
47 | For $n\ge1$ 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 $360.$ | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-12', 'Year': 2019, 'Problem Number': 12, 'Part': 'II'} | null | null | null |
504 | Regular 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 $\overline{PA_3},\overline{PA_4},$ and the minor arc $\widehat{A_3A_4}$ of the circle has area $\tfrac{1}{9}.$ There is a positive integer $n$ such that the area of the region bounded by $\overline{PA_6},\overline{PA_7},$ and the minor arc $\widehat{A_6A_7}$ of the circle is equal to $\tfrac{1}{8}-\tfrac{\sqrt2}{n}.$ Find $n.$ | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-13', 'Year': 2019, 'Problem Number': 13, 'Part': 'II'} | null | null | null |
71 | Find 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. | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-14', 'Year': 2019, 'Problem Number': 14, 'Part': 'II'} | null | null | null |
574 | In 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 $QY=15$ . The value of $AB\cdot AC$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$ . | math | qq8933/AIME_1983_2024 | {'ID': '2019-II-15', 'Year': 2019, 'Problem Number': 15, 'Part': 'II'} | null | null | null |
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