US-EGMO-TST 2020

Let $ABC$ be a triangle and let $P$ be a point not lying on any of the three lines $AB$ , $BC$ , or $CA$ . Distinct points $D$ , $E$ , and $F$ lie on lines $BC$ , $AC$ , and $AB$ , respectively, such that $\overline{DE}\parallel \overline{CP}$ and $\overline{DF}\parallel \overline{BP}$ . Show that there exists a point $Q$ on the circumcircle of $\triangle AEF$ such that $\triangle BAQ$ is similar to $\triangle PAC$ .

Andrew Gu

Let $ABC$ be a triangle. Distinct points $D$ , $E$ , $F$ lie on sides $BC$ , $AC$ , and $AB$ , respectively, such that quadrilaterals $ABDE$ and $ACDF$ are cyclic. Line $AD$ meets the circumcircle of $\triangle ABC$ again at $P$ . Let $Q$ denote the reflection of $P$ across $BC$ . Show that $Q$ lies on the circumcircle of $\triangle AEF$ .

Proposed by Ankan Bhattacharya

Let $G = (V, E)$ be a finite simple graph on $n$ vertices. An edge $e$ of $G$ is called a bottleneck if one can partition $V$ into two disjoint sets $A$ and $B$ such that

• at most $100$ edges of $G$ have one endpoint in $A$ and one endpoint in $B$ ; and
• the edge $e$ is one such edge (meaning the edge $e$ also has one endpoint in $A$ and one endpoint in $B$ ).

Prove that at most $100n$ edges of $G$ are bottlenecks.

Proposed by Yang Liu

Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$ , $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$ . Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.