Let $n$ be a positive integer. Define a sequence by setting $a_1 = n$ and, for each $k > 1$, letting $a_k$ be the unique integer in the range $0 \leq a_k \leq k-1$ for which $a_1 + a_2 + \dots + a_k$ is divisible by $k$. (For instance, when $n = 9$ the obtained sequence is $9,1,2,0,3,3,3,\dots$.) Prove that for any $n$ the sequence $a_1,a_2,\ldots$ eventually becomes constant.
Proposed by Sam Vandervelde; source discussion: https://aops.com/community/p825490
Decide whether it possible to cover all lattice points in $\mathbb{R}^2$ by an (infinite) family of disks whose interiors are disjoint such that the radius of each disk is at least $5$.
Proposed by Gregory Galperin; source discussion: https://aops.com/community/p825495
Let $S$ be a set containing $n^2+n-1$ elements. Suppose that the $n$-element subsets of $S$ are partitioned into two classes. Prove that there are at least $n$ pairwise disjoint sets in the same class.
Proposed by Andras Gyarfas; source discussion: https://aops.com/community/p825499
An animal with $n$ cells is a connected figure consisting of $n$ equal-sized square cells (equivalently, a polyomino with $n$ cells). A dinosaur is an animal with at least $2007$ cells. It is said to be primitive if its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
Proposed by Reid Barton; source discussion: https://aops.com/community/p825505
Let $ABC$ be an acute triangle with $\omega$, $S$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_{A}$ is tangent internally to $S$ at $A$ and tangent externally to $\omega$. Circle $S_{A}$ is tangent internally to $S$ at $A$ and tangent internally to $\omega$. Let $P_A$ and $Q_A$ denote the centers of $\omega_A$ and $S_A$, respectively. Define points $P_B$, $Q_B$, $P_C$, $Q_C$ analogously. Prove that