There are $n \ge 3$ positive real numbers $a_1, a_2, \dots, a_n$. For each $1 \le i \le n$ we let $b_i = \frac{a_{i-1} + a_{i+1}}{a_i}$ (here we define $a_0$ to be $a_n$ and $a_{n+1}$ to be $a_1$). Assume that for all $i$ and $j$ in the range $1$ to $n$, we have $a_i \le a_j$ if and only if $b_i \le b_j$. Prove that $a_1 = a_2 = \dots = a_n$.

We are given an acute triangle $ABC$. Let $D$ be the point on its circumcircle such that $AD$ is a diameter. Suppose that points $K$ and $L$ lie on segments $AB$ and $AC$, respectively, and that $DK$ and $DL$ are tangent to circle $AKL$. Show that line $KL$ passes through the orthocenter of triangle $ABC$.

Let $k$ be a positive integer. Lexi has a dictionary $\mathbb{D}$ consisting of some $k$-letter strings containing only the letters $A$ and $B$. Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathbb{D}$ when read from top-to-bottom and each row contains a string from $\mathbb{D}$ when read from left-to-right. What is the smallest integer $m$ such that if $\mathbb{D}$ contains at least $m$ different strings, then Lexi can fill her grid in this manner, no matter what strings are in $\mathbb{D}$?

Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise. Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.

We are given a positive integer $s \ge 2$. For each positive integer $k$, we define its twist $k’$ as follows: write $k$ as $as+b$, where $a, b$ are non-negative integers and $b < s$, then $k’ = bs+a$. For the positive integer $n$, consider the infinite sequence $d_1, d_2, \dots$ where $d_1=n$ and $d_{i+1}$ is the twist of $d_i$ for each positive integer $i$. Prove that this sequence contains $1$ if and only if the remainder when $n$ is divided by $s^2-1$ is either $1$ or $s$.

Let $ABC$ be a triangle with circumcircle $\Omega$. Let $S_b$ and $S_c$ respectively denote the midpoints of the arcs $AC$ and $AB$ that do not contain the third vertex. Let $N_a$ denote the midpoint of arc $BAC$ (the arc $BC$ including $A$). Let $I$ be the incenter of $ABC$. Let $\omega_b$ be the circle that is tangent to $AB$ and internally tangent to $\Omega$ at $S_b$, and let $\omega_c$ be the circle that is tangent to $AC$ and internally tangent to $\Omega$ at $S_c$. Show that the line $IN_a$, and the lines through the intersections of $\omega_b$ and $\omega_c$, meet on $\Omega$.