VMO 2015

For $a,n\in\mathbb{Z}^+$ , consider the following equation: $$a^2x+6ay+36z=n (1)$$ where $x,y,z\in\mathbb{N}$ .

a) Find all $a$ such that for all $n\geq 250$ , $(1)$ always has natural roots $(x,y,z)$ .

b) Given that $a>1$ and $\gcd (a,6)=1$ . Find the greatest value of $n$ in terms of $a$ such that $(1)$ doesn't have natural root $(x,y,z)$ .

Given a circumcircle $(O)$ and two fixed points $B,C$ on $(O)$ . $BC$ is not the diameter of $(O)$ . A point $A$ varies on $(O)$ such that $ABC$ is an acute triangle. $E,F$ is the foot of the altitude from $B,C$ respectively of $ABC$ . $(I)$ is a variable circumcircle going through $E$ and $F$ with center $I$ .

a) Assume that $(I)$ touches $BC$ at $D$ . Probe that $\frac{DB}{DC}=\sqrt{\frac{\cot B}{\cot C}}$ .

b) Assume $(I)$ intersects $BC$ at $M$ and $N$ . Let $H$ be the orthocenter and $P,Q$ be the intersections of $(I)$ and $(HBC)$ . The circumcircle $(K)$ going through $P,Q$ and touches $(O)$ at $T$ ( $T$ is on the same side with $A$ wrt $PQ$ ). Prove that the interior angle bisector of $\angle{MTN}$ passes through a fixed point.