Raashan, Sylvia, and Ted each start with one dollar. Every $15$ seconds a bell rings. Each player who currently has money simultaneously and independently chooses one of the other two players at random and gives that player one dollar.
What is the probability that after $2019$ rings each player has exactly one dollar?
Let $ABCDEF$ be a regular hexagon with side length $1.$ Denote by $X,$ $Y,$ and $Z$ the midpoints of sides $AB,$ $CD,$ and $EF,$ respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle ACE$ and $\triangle XYZ?$
Three real numbers $x,y,z$ are chosen independently and uniformly from $[0,n]$, where $n$ is a positive integer. Let $P$ be the probability that every pair is at distance at least $1$. If $P\gt \dfrac12$, what is the least possible $n$?
Let $T_1$ be a triangle with side lengths $2011,$ $2012,$ and $2013.$ For $n \ge 1,$ if $T_n = \triangle ABC$ and $D,$ $E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,$ $BC,$ and $AC,$ respectively, then $T_{n+1}$ is a triangle with side lengths $AD,$ $BE,$ and $CF,$ if it exists. What is the perimeter of the last triangle in the sequence $( T_n )?$
A bug starts at a vertex of a cube. At each vertex it independently chooses one of the three incident edges with equal probability and traverses it. What is the probability that after seven moves it has visited all eight vertices exactly once?
Eleven lily pads in a row are labeled $0$ through $10$. A frog starts on pad $1$. From pad $N$, where $0\lt N\lt 10$, it jumps to $N-1$ with probability $\dfrac N{10}$ and to $N+1$ with probability $1-\dfrac N{10}$, with each choice depending only on its current position.
At pad $0$ it is eaten; at pad $10$ it escapes permanently. What is its probability of escape?
In trapezoid $ABCD,$ $\overline{AB}$ and $\overline{CD}$ are perpendicular to $\overline{AD},$ with $AB+CD=BC,$ $AB\lt CD,$ and $AD=7.$ What is $AB\cdot CD?$
Jason rolls three fair six-sided dice. After seeing them, he chooses a subset to reroll, possibly none or all three. He wins if the resulting sum is exactly $7$, and always maximizes his winning probability. What is the probability that he rerolls exactly two dice?
Circles $A,$ $B,$ and $C$ are externally tangent to each other and internally tangent to circle $D.$ Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D.$ What is the radius of circle $B?$
In $\triangle ABC,$ $AB = 86,$ and $AC=97.$ A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X.$ Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is $BC?$
In triangle $\triangle ABC,$ $AB=13,$ $BC=14,$ and $CA=15.$ Distinct points $D,$ $E,$ and $F$ lie on segments $\overline{BC},$ $\overline{CA},$ and $\overline{DE},$ respectively, such that $\overline{AD}\perp\overline{BC},$ $\overline{DE}\perp\overline{AC},$ and $\overline{AF}\perp\overline{BF}.$ The length of segment $\overline{DF}$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. What is $m+n?$
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\%$ on every pair of shoes. Carlos also knew that he had to pay a $7.5\%$ sales tax on the discounted price. He had $43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
Define $x~\diamondsuit~ y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of $$
(1~\diamondsuit~(2~\diamondsuit~3))-((1~\diamondsuit~2)~\diamondsuit~3)?
$$
Suppose $a$ and $b$ are real numbers. When the polynomial $x^3 + x^2 + ax + b$ is divided by $x - 1,$ the remainder is $4.$ When the polynomial is divided by $x - 2,$ the remainder is $6.$ What is $b - a?$
Let $L_1 = 1,$ $L_2 = 3,$ and $L_{n+2} = L_{n+1} + L_n$ for $n \ge 1.$ How many terms in the sequence $L_1,$ $L_2,$ $L_3,$ $\ldots,$ $L_{2023}$ are even?
The value of the two-digit number $\underline{a}\,\underline{b}$ in base seven equals the value of the two-digit number $\underline{b}\,\underline{a}$ in base nine. What is $a + b?$
A rectangle is partitioned into $5$ regions as shown. Each region is to be painted a solid color—red, orange, yellow, blue, or green—so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
