MEMO 2015

Let $ABC$ be an acute triangle with $AB>AC$ . Prove that there exists a point $D$ with the following property: whenever two distinct points $X$ and $Y$ lie in the interior of $ABC$ such that the points $B$ , $C$ , $X$ , and $Y$ lie on a circle and $$\angle AXB-\angle ACB=\angle CYA-\angle CBA$$ holds, the line $XY$ passes through $D$ .

Let $I$ be the incentre of triangle $ABC$ with $AB>AC$ and let the line $AI$ intersect the side $BC$ at $D$ . Suppose that point $P$ lies on the segment $BC$ and satisfies $PI=PD$ . Further, let $J$ be the point obtained by reflecting $I$ over the perpendicular bisector of $BC$ , and let $Q$ be the other intersection of the circumcircles of the triangles $ABC$ and $APD$ . Prove that $\angle BAQ=\angle CAJ$ .

Find all pairs of positive integers $(a,b)$ such that $$a!+b!=a^b + b^a.$$

Let $n\ge 2$ be an integer. Determine the number of positive integers $m$ such that $m\le n$ and $m^2+1$ is divisible by $n$ .