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 $m + n.$
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 $MO.$
There are positive integers $x$ and $y$ that satisfy the system of equations $$
\log_{10} x + 2\log_{10}(\gcd(x, y)) = 60
$$ $$
\begin{aligned} &\log_{10} y \\ &\quad {}+ 2\log_{10}(\operatorname{lcm}(x, y)) = 570. \end{aligned}
$$ 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 $3m + 2n.$
Let $x$ be a real number such that $\sin^{10} x + \cos^{10} x = \frac{11}{36}.$ Then $\sin^{12} x + \cos^{12} x = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
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 $\tau(n) + \tau(n + 1) = 7.$
For distinct complex numbers $z_1,$ $z_2,$ $\ldots,$ $z_{673},$ the polynomial $$
\begin{gathered} (x - z_1)^3 (x - z_2)^3 \\ \cdots (x - z_{673})^3 \end{gathered}
$$ 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 $$
\left| \sum_{1 \le j \lt k \le 673} z_j z_k \right|
$$ can be expressed in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
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 $\triangle ABC.$
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 $m + \sqrt{n} + 11\mathrm{i}.$ Find $m + n.$
Triangle $ABC$ has side lengths $AB = 4,$ $BC = 5,$ and $CA = 6.$ Points $D$ and $E$ are on ray $AB$ with $AB \lt AD \lt AE.$ 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 $\frac{a + b\sqrt{c}}{d},$ 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 $a + b + c + d.$
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 $PQ^2 = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$