Let $n, k \ge 2$ be positive integers and let $a_1$, $a_2$, $a_3$, …, $a_k$ be distinct integers in the set $\left\{ 1,2,\dots,n \right\}$ such that $n$ divides $a_i(a_{i+1} - 1)$ for $i = 1,2,\dots,k-1$. Prove that $n$ does not divide $a_k(a_1 - 1)$.
Let $ABC$ be a triangle with circumcenter $O$. The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$ respectively. Let $K$, $L$, $M$ be the midpoints of $\overline{BP}$, $\overline{CQ}$, $\overline{PQ}$. Suppose that $\overline{PQ}$ is tangent to the circumcircle of $\triangle KLM$. Prove that $OP = OQ$.
Suppose that $s_1,s_2,s_3, \dotsc$ is a strictly increasing sequence of positive integers such that the sub-sequences $s_{s_1}$, $s_{s_2}$, $s_{s_3}$, … and $s_{s_1 + 1}$, $s_{s_2 + 1}$, $s_{s_3 + 1}$, … are both arithmetic progressions. Prove that the sequence $s_1$, $s_2$, $s_3$, … is itself an arithmetic progression.
Let $ABC$ be a triangle with $AB = AC$. The angle bisectors of $\angle CAB$ and $\angle ABC$ meet the sides $BC$ and $CA$ at $D$ and $E$, respectively. Let $K$ be the incenter of triangle $ADC$. Suppose that $\angle BEK = 45^\circ$. Find all possible values of $\angle CAB$.
Find all functions $f \colon \mathbb{Z}_{\gt 0} \to \mathbb{Z}_{\gt 0}$ such that for positive integers $a$ and $b$, the numbers $$
a, \qquad f(b), \qquad f(b+f(a)-1)
$$ are the sides of a non-degenerate triangle. A triangle is non-degenerate if its vertices are not collinear.
Let $a_1$, $a_2$, …, $a_n$ be distinct positive integers and let $M$ be a set of $n-1$ positive integers not containing $s = a_1 + \dots + a_n$. A grasshopper is to jump along the real axis, starting at the point $0$ and making $n$ jumps to the right with lengths $a_1$, $a_2$, …, $a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M$.