FKMO 2016

In a acute triangle $\triangle ABC$ , denote $D, E$ as the foot of the perpendicular from $B$ to $AC$ and $C$ to $AB$ . Denote the reflection of $E$ with respect to $AC, BC$ as $S, T$ . The circumcircle of $\triangle CST$ hits $AC$ at point $X (\not= C)$ . Denote the circumcenter of $\triangle CST$ as $O$ . Prove that $XO \perp DE$ .

Two integers $n, k$ satisfies $n \ge 2$ and $k \ge \frac{5}{2}n-1$ . Prove that whichever $k$ lattice points with $x$ and $y$ coordinate no less than $1$ and no more than $n$ we pick, there must be a circle passing through at least four of these points.

Prove that for all rationals $x,y$ , $x-\frac{1}{x}+y-\frac{1}{y}=4$ is not true.

If $x,y,z$ satisfies $x^2+y^2+z^2=1$ , find the maximum possible value of $$(x^2-yz)(y^2-zx)(z^2-xy)$$

Let $U$ be a set of $m$ triangles. Prove that there exists a subset $W$ of $U$ which satisfies the following.

(i). The number of triangles in $W$ is at least $0.45m^{\frac{4}{5}}$ (ii) There are no points $A, B, C, D, E, F$ such that triangles $ABC$ , $BCD$ , $CDE$ , $DEF$ , $EFA$ , $FAB$ are all in $W$ .