ELMO 2021

In $\triangle ABC$ , points $P$ and $Q$ lie on sides $AB$ and $AC$ , respectively, such that the circumcircle of $\triangle APQ$ is tangent to $BC$ at $D$ . Let $E$ lie on side $BC$ such that $BD = EC$ . Line $DP$ intersects the circumcircle of $\triangle CDQ$ again at $X$ , and line $DQ$ intersects the circumcircle of $\triangle BDP$ again at $Y$ . Prove that $D$ , $E$ , $X$ , and $Y$ are concyclic.

Let $n > 1$ be an integer and let $a_1, a_2, \ldots, a_n$ be integers such that $n \mid a_i-i$ for all integers $1 \leq i \leq n$ . Prove there exists an infinite sequence $b_1,b_2, \ldots$ such that

• $b_k\in\{a_1,a_2,\ldots, a_n\}$ for all positive integers $k$ , and
• $\sum\limits_{k=1}^{\infty}\frac{b_k}{n^k}$ is an integer.

Each cell of a $100\times 100$ grid is colored with one of $101$ colors. A cell is diverse if, among the $199$ cells in its row or column, every color appears at least once. Determine the maximum possible number of diverse cells.

The set of positive integers is partitioned into $n$ disjoint infinite arithmetic progressions $S_1, S_2, \ldots, S_n$ with common differences $d_1, d_2, \ldots, d_n$ , respectively. Prove that there exists exactly one index $1\leq i \leq n$ such that$$\frac{1}{d_i}\prod_{j=1}^n d_j \in S_i.$$

Let $n$ and $k$ be positive integers. Two infinite sequences $\{s_i\}_{i\geq 1}$ and $\{t_i\}_{i\geq 1}$ are equivalent if, for all positive integers $i$ and $j$ , $s_i = s_j$ if and only if $t_i = t_j$ . A sequence $\{r_i\}_{i\geq 1}$ has equi-period $k$ if $r_1, r_2, \ldots$ and $r_{k+1}, r_{k+2}, \ldots$ are equivalent.

Suppose $M$ infinite sequences with equi-period $k$ whose terms are in the set $\{1, \ldots, n\}$ can be chosen such that no two chosen sequences are equivalent to each other. Determine the largest possible value of $M$ in terms of $n$ and $k$ .

In $\triangle ABC$ , points $D$ , $E$ , and $F$ lie on sides $BC$ , $CA$ , and $AB$ , respectively, such that each of the quadrilaterals $AFDE$ , $BDEF$ , and $CEFD$ has an incircle. Prove that the inradius of $\triangle ABC$ is twice the inradius of $\triangle DEF$ .