FKMO 2018

Find all integers of the form $\frac{m-6n}{m+2n}$ where $m,n$ are nonzero rational numbers satisfying $m^3=(27n^2+1)(m+2n)$ .

Triangle $ABC$ satisfies $\angle ABC < \angle BCA < \angle CAB < 90^{\circ}$ . $O$ is the circumcenter of triangle $ABC$ , and $K$ is the reflection of $O$ in $BC$ . $D,E$ is the foot of perpendicular line from $K$ to line $AB$ , $AC$ , respectively. Line $DE$ meets $BC$ at $P$ , and a circle with diameter $AK$ meets the circumcircle of triangle $ABC$ at $Q(\neq A)$ . If $PQ$ cuts the perpendicular bisector of $BC$ at $S$ , then prove that $S$ lies on the circle with diameter $AK$ .

For 31 years, n (>6) tennis players have records of wins. It turns out that for every two players, there is a third player who has won over them before. Prove that for every integer $k,l$ such that $2^{2^k+1}-1>n, 1<l<2k+1$ , there exist $l$ players ( $A_1, A_2, ... , A_l$ ) such that every player $A_{i+1}$ won over $A_i$ . ( $A_{l+1}$ is same as $A_1$ )

Triangle $ABC$ satisfies $\angle C=90^{\circ}$ . A circle passing $A,B$ meets segment $AC$ at $G(\neq A,C)$ and it meets segment $BC$ at point $D(\neq B)$ . Segment $AD$ cuts segment $BG$ at $H$ , and let $l$ , the perpendicular bisector of segment $AD$ , cuts the perpendicular bisector of segment $AB$ at point $E$ . A line passing $D$ is perpendicular to $DE$ and cuts $l$ at point $F$ . If the circumcircle of triangle $CFH$ cuts $AC$ , $BC$ at $P(\neq C),Q(\neq C)$ respectively, then prove that $PQ$ is perpendicular to $FH$ .

Twenty ants live on the faces of an icosahedron, one ant on each side, where the icosahedron have each side with length 1. Each ant moves in a counterclockwise direction on each face, along the side/edges. The speed of each ant must be no less than 1 always. Also, if two ants meet, they should meet at the vertex of the icosahedron. If five ants meet at the same time at a vertex, we call that a collision. Can the ants move forever, in a way that no collision occurs?