IMAC 2013

Show that in any set of three distinct integers there are two of them, say $a$ and $b$ such that the number $a^5b^3-a^3b^5$ is a multiple of $10$ .

For all positive integer $n$ , we consider the number $$a_n =4^{6^n}+1943$$ Prove that $a_n$ is dividible by $2013$ for all $n\ge 1$ , and find all values of $n$ for which $a_n - 207$ is the cube of a positive integer.

Let $ABC$ be a triangle with $\angle ABC=120^o$ and triangle bisectors $(AA_1),(BB_1),(CC_1)$ , respectively. $B_1F \perp A_1C_1$ , where $F\in (A_1C_1)$ . Let $R,I$ and $S$ be the centers of the circles which are inscribed in triangles $C_1B_1F,C_1B_1A_1, A_1B_1F$ , and $B_1S\cap A_1C_1=\{Q\}$ . Show that $R,I,S,Q$ are on the same circle.