IMAC 2012

Let $a_1,a_2,..., a_n$ be different integers and let $(b_1,b_2,..., b_n),(c_1,c_2,..., c_n)$ be two of their permutations, different from the identity. Prove that $$(|a_1-b_1|+|a_2-b_2|+...+|a_n-b_n| , |a_1-c_1|+|a_2-c_2|+...+|a_n-c_n| ) \ge 2$$ where $(x,y)$ denotes the greatest common divisor of the numbers $x,y$

Circles $k_1,k_2$ intersect at $B,C$ such that $BC$ is diameter of $k_1$ .Tangent of $k_1$ at $C$ touches $k_2$ for the second time at $A$ .Line $AB$ intersects $k_1$ at $E$ different from $B$ , and line $CE$ intersects $k_2$ at F different from $C$ . An arbitrary line through $E$ intersects segment $AF$ at $H$ and $k_1$ for the second time at $G$ .If $BG$ and $AC$ intersect at $D$ , prove $CH//DF$ .

Find all functions $f:Q^+ \to Q^+$ such that for any $x,y \in Q^+$ : $$y=\frac{1}{2}\left[f\left(x+\frac{y}{x}\right)- \left(f(x)+\frac{f(y)}{f(x)}\right)\right]$$