Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that: $$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.

Find the number of ways to color $n \times m$ board with white and black colors such that any $2 \times 2$ square contains the same number of black and white cells.

For each $n \in N$ let $S(n)$ be the sum of all numbers in the set {1,2,3,…,n} which are relatively prime to $n$. a. Show that $2S(n) $ is not aperfect square for any $n$. b. Given positive integers $m,n$ with odd n, show that the equation $2S(x)=y^n$ has at least one solution $(x,y)$ among positive integers such that $m|x$.

Let $ABC$ be a triangle with incircle touching $BC, CA, AB$ at $D, E, F,$ respectively. Let $O$ and $M$ be its circumcenter and midpoint of $BC.$ Suppose that circumcircles of $AEF$ and $ABC$ intersect at $X$ for the second time. Assume $Y \neq X$ is on the circumcircle of $ABC$ such that $OMXY$ is cyclic. Prove that circumcenter of $DXY$ lies on $BC.$ Proposed by tenplusten.