SILK-ROAD 2005

Find all $(m,n) \in \mathbb{Z}^2$ that we can color each unit square of $m \times n$ with the colors black and white that for each unit square number of unit squares that have the same color with it and have at least one common vertex (including itself) is even.

Assume $A,B,C$ are three collinear points that $B \in [AC]$ . Suppose $AA'$ and $BB'$ are to parrallel lines that $A'$ , $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$ , $A'$ and $C$ . Also $O_2$ is circumcenter of circle passing through $B$ , $B'$ and $C$ . If area of $A'CB'$ is equal to area of $O_1CO_2$ , then find all possible values for $\angle CAA'$

Suppose $\{a(n) \}_{n=1}^{\infty}$ is a sequence that: $$a(n) =a(a(n-1))+a(n-a(n-1)) \ \ \ \forall \ n \geq 3$$ and $a(1)=a(2)=1$ .

Prove that for each $n \geq 1$ , $a(2n) \leq 2a(n)$ .