There are $n$ assassins numbered from $1$ to $n,$ and all assasins are initially alive. The assassins play a game in which they take turns in increasing order of number, with assassin $1$ getting the first turn, then assassin $2$, etc., with the order repeating after assassin $n$ has gone; if an assassin is dead when their turn comes up, then their turn is skipped and it goes to the next assassin in line. On each assassin’s turn, they can choose to either kill the assassin who would otherwise move next or to do nothing. Each assassin will kill on their turn unless the only option for guaranteeing their own survival is to do nothing. If there are $2023$ assassins at the start of the game, after an entire round of turns in which no one kills, how many assassins must remain?

The number $2019$ is written on a blackboard. Every minute, if the number $a$ is written on the board, Evan erases it and replaces it with a number chosen from the set $$ \left\{ 0, 1, 2, \ldots, \left\lceil 2.01 a \right\rceil \right\} $$ uniformly at random. Is there an integer $N$ such that the board reads $0$ after $N$ steps with at least $99\%$ probability?

A chocolate bar in the shape of an equilateral triangle with side of the length $n$, consists of triangular chips with sides of the length $1$, parallel to sides of the bar. Two players take turns eating up the chocolate. Each player breaks off a triangular piece (along one of the lines), eats it up and passes leftovers to the other player (as long as bar contains more than one chip, the player is not allowed to eat it completely). A player who has no move or leaves exactly one chip to the opponent, loses. For each $n$, find who has a winning strategy.

Triangle $ ABC$ has side lengths $ AB \equal{} 5$, $ BC \equal{} 6$, and $ AC \equal{} 7$. Two bugs start simultaneously from $ A$ and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point $ D$. What is $ BD$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Serena has written 20 copies of the number 1 on a board. In a move, she is allowed to $\quad *$ erase two of the numbers and replace them with their sum, or $\quad *$ erase one number and replace it with its reciprocal. Whenever a fraction appears on the board, Serena writes it in simplest form. Prove that Serena can never write a fraction less than 1 whose numerator is over 9000, regardless of the number of moves she makes.

For $x,y,z > 0$ and $xyz=1$, prove that \[\frac{x^{9}+y^{9}}{x^{6}+x^{3}y^{3}+y^{6}}+\frac{x^{9}+z^{9}}{x^{6}+x^{3}z^{3}+z^{6}}+\frac{y^{9}+z^{9}}{y^{6}+y^{3}z^{3}+z^{6}}\geq 2\]

Define mutually externally tangent circles $\omega_1$, $\omega_2$, and $\omega_3$. Let $\omega_1$ and $\omega_2$ be tangent at $P$. The common external tangents of $\omega_1$ and $\omega_2$ meet at $Q$. Let $O$ be the center of $\omega_3$. If $QP = 420$ and $QO = 427$, find the radius of $\omega_3$. [i]Proposed by Tanishq Pauskar and Mahith Gottipati[/i]

Find the maximum number of points on a sphere of radius $1$ in $\mathbb{R}^n$ such that the distance between any two of these points is strictly greater than $\sqrt{2}$.

A regular hexagon is inscribed in a circle of radius $ 10$ inches. Its area is: $ \textbf{(A)}\ 150\sqrt{3} \text{ sq. in.} \qquad \textbf{(B)}\ \text{150 sq. in.} \qquad \textbf{(C)}\ 25\sqrt{3} \text{ sq. in.} \qquad \textbf{(D)}\ \text{600 sq. in.} \qquad \textbf{(E)}\ 300\sqrt{3} \text{ sq. in.}$

Let ${n}$ be a fixed positive integer. ${A}$ and ${B}$ play the following game: $2023$ coins marked $1, 2, \dots, 2023$ lie on a circle (the marks are considered in module $2023$) and each coin has two sides. Initially, all coins are head up and ${A}$'s goal is to make as many coins with tail up. In each operation, ${A}$ choose two coins marked ${k}$ and $k+3$ with head up (if ${A}$ can't choose, the game ends) and ${B}$ choose a coin marked $k+1$ or $k+2$ and flip it. If at some moment there are ${n}$ coins with tail up, ${A}$ wins. Find the largest ${n}$ such that ${A}$ has a winning strategy.

Let $\{a_n\}$ be a nonnegative real sequence. Define $$X_k = \sum_{i=1}^{2^k}a_i, Y_k = \sum_{i=1}^{2^k}\left\lfloor \frac{2^k}{i}\right\rfloor a_i, k=0,1,2,...$$ Prove that $X_n\le Y_n - \sum_{i=0}^{n-1} Y_i \le \sum_{i=0}^n X_i$ for all positive integer $n$. Here $\lfloor\alpha\rfloor$ denotes the largest integer that does not exceed $\alpha$.

Determine the maximal value of $k$ such that the inequality $$\left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right) \le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left( \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right)$$ holds for all positive reals $a, b, c$.

In a non-isosceles triangle $ABC$,$O$ and $I$ are circumcenter and incenter,respectively.$B^\prime$ is reflection of $B$ with respect to $OI$ and lies inside the angle $ABI$.Prove that the tangents to circumcirle of $\triangle BB^\prime I$ at $B^\prime$,$I$ intersect on $AC$. (A. Kuznetsov)

Let $z_1,z_2,...,z_n$ be complex numbers satisfying $|z_i - 1| \leq r$ for some $r$ in $(0,1)$. Show that \[ \left | \sum_{i=1}^n z_i \right | \cdot \left | \sum_{i=1}^n \frac{1}{z_i} \right | \geq n^2(1-r^2).\]

An arbitrary point $D$ is marked on the hypotenuse $AB$ of a right triangle $ABC$. The circle circumscribed around the triangle $ACD$ intersects the line $BC$ at the point $E$ for the second time, and the circle circumscribed around the triangle $BCD$ intersects the line $AC$ for the second time at the point $F$. Prove that the line $EF$ passes through the point $D$.

Given the circumradius $R$ of a triangle, a side length $c$, and the ratio $a/b$ of the other two side lengths, determine all three sides and angles of this triangle.

Let $N = 2014! + 2015! + 2016! + \dots + 9999!$. How many zeros are at the end of the decimal representation of $N$? [i]Proposed by Evan Chen[/i]

For any positive integer n, let $\varphi(n)$ be the number of integers in the set $\{1, 2, \ldots , n\}$ whose greatest common divisor with $n$ is 1. Determine the maximum value of $\frac{n}{\varphi(n)}$ for $n$ in the set $\{2, \ldots, 1000\}$ and all values of $n$ for which this maximum is attained.

In the figure, the rectangle is formed by $4$ smaller equal rectangles. If we count the total number of rectangles in the figure we find $10$. How many rectangles in total will there be in a rectangle that is formed by $n$ smaller equal rectangles?

The roots of the two monic square trinomials are negative integers, and one of these roots is common. Can the values of these trinomials at some positive integer point equal 19 and 98?

A sequence of positive real numbers $a_1, a_2, a_3, ... $ satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. A sequence $b_1, b_2, b_3, ...$ is defined by equations $b_1 = a_1$ , $b_n = a_n + (b_1 + b_3 + ...+ b_{n-1})$ for even $n > 1$ , $b_n = a_n + (b_2 + b_4 + ... +b_{n-1})$ for odd $n > 1$. Prove that if $n\ge 3$, then $\frac13 < \frac{b_n}{n \cdot a_n} < 1$

Quadrilateral $ABCD$ with $AC = 800$ is inscribed in a circle, and $E, W, X, Y, Z$ are the midpoints of segments $BD$, $AB$, $BC$, $CD$, $DA$, respectively. If the circumcenters of $EW Z$ and $EXY$ are $O_1$ and $O_2$, respectively, determine $O_1O_2$.

In a right triangle $ABC$, M is a point on the hypotenuse $BC$ and $P$ and $Q$ the projections of $M$ on $AB$ and $AC$ respectively. Prove that for no such point $M$ do the triangles $BPM, MQC$ and the rectangle $AQMP$ have the same area.

Consider the set $A=\{1,2,...,49\}$. We partitionate $A$ into three subsets. Prove that there exist a set from these subsets containing three distincts elements $a,b,c$ such that $a+b=c$

Find all positive integers $m,n$ with $m \leq 2n$ that solve the equation \[ m \cdot \binom{2n}{n} = \binom{m^2}{2}. \] [i](German MO 2016 - Problem 4)[/i]