FKMO 2015

Find all functions $f: R \rightarrow R$ such that $f(x^{2015} + (f(y))^{2015}) = (f(x))^{2015} + y^{2015}$ holds for all reals $x, y$

In a triangle $\triangle ABC$ with incenter $I$ , the incircle meets lines $BC, CA, AB$ at $D, E, F$ respectively. Define the circumcenter of $\triangle IAB$ and $\triangle IAC$ $O_1$ and $O_2$ respectively. Let the two intersections of the circumcircle of $\triangle ABC$ and line $EF$ be $P, Q$ . Prove that the circumcenter of $\triangle DPQ$ lies on the line $O_1O_2$ .

There are at least $3$ subway stations in a city. In this city, there exists a route that passes through more than $L$ subway stations, without revisiting. Subways run both ways, which means that if you can go from subway station A to B, you can also go from B to A. Prove that at least one of the two holds. $\text{(i)}$ . There exists three subway stations $A$ , $B$ , $C$ such that there does not exist a route from $A$ to $B$ which doesn't pass through $C$ . $\text{(ii)}$ . There is a cycle passing through at least $\lfloor \sqrt{2L} \rfloor$ stations, without revisiting a same station more than once.

$\triangle ABC$ is an acute triangle and its orthocenter is $H$ . The circumcircle of $\triangle ABH$ intersects line $BC$ at $D$ . Lines $DH$ and $AC$ meets at $P$ , and the circumcenter of $\triangle ADP$ is $Q$ . Prove that the circumcenter of $\triangle ABH$ lies on the circumcircle of $\triangle BDQ$ .

For a fixed positive integer $k$ , there are two sequences $A_n$ and $B_n$ . They are defined inductively, by the following recurrences. $A_1 = k$ , $A_2 = k$ , $A_{n+2} = A_{n}A_{n+1}$ $B_1 = 1$ , $B_2 = k$ , $B_{n+2} = \frac{B^3_{n+1}+1}{B_{n}}$ Prove that for all positive integers $n$ , $A_{2n}B_{n+3}$ is an integer.

There are $2015$ distinct circles in a plane, with radius $1$ . Prove that you can select $27$ circles, which form a set $C$ , which satisfy the following.

For two arbitrary circles in $C$ , they intersect with each other or For two arbitrary circles in $C$ , they don't intersect with each other.