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 $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
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 $\frac{1}{2}$ and independently of other jumps. The probability that the frog visits pad $7$ is $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p + q.$
Find the number of $7$-tuples of positive integers $(a, b, c, d, e, f, g)$ that satisfy the following system of equations: $$
abc = 70,
$$ $$
cde = 71,
$$ $$
efg = 72.
$$
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 $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
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 $1000.$
In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b,$ for some fixed $b \ge 2.$ 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 $x \gt 1.$ Find $b.$
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 $\ell_C.$
The polynomial $f(z) = az^{2018} + bz^{2017} + cz^{2016}$ has real coefficients not exceeding $2019,$ and $f\left(\frac{1 + \sqrt{3}i}{2}\right) = 2015 + 2019\sqrt{3}i.$ Find the remainder when $f(1)$ is divided by $1000.$
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 $\frac{S}{20}.$
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 $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p + q.$
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 $AK = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
For $n \ge 1$ call a finite sequence $(a_1, a_2, \ldots, a_n)$ of positive integers *progressive* if $a_i \lt a_{i+1}$ 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 $360.$
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 $\overset{\frown}{A_1A_2}$ of the circle has area $\frac{1}{7},$ while the region bounded by $\overline{PA_3},$ $\overline{PA_4},$ and the minor arc $\overset{\frown}{A_3A_4}$ of the circle has area $\frac{1}{9}.$ 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 $\overset{\frown}{A_6A_7}$ of the circle is equal to $\frac{1}{8} - \frac{\sqrt{2}}{n}.$ Find $n.$
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.
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 $m + n.$
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 $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
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 $ABCDEF.$
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 $1000.$
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 $\frac{1}{3},$ independently of its previous moves. The probability that it will hit the coordinate axes at $(0, 0)$ is $\frac{m}{3^n},$ where $m$ and $n$ are positive integers, and $m$ is not divisible by $3.$ Find $m + n.$