RS-SMO 2011

Let $n \ge 2$ be integer. Let $a_0$ , $a_1$ , ... $a_n$ be sequence of positive reals such that: $(a_{k-1}+a_k)(a_k+a_{k+1})=a_{k-1}-a_{k+1}$ , for $k=1, 2, ..., n-1$ . Prove $a_n< \frac{1}{n-1}$ .

Let $n$ be an odd positive integer such that both $\phi(n)$ and $\phi (n+1)$ are powers of two. Prove $n+1$ is power of two or $n=5$ .

Let $H$ be orthocenter and $O$ circumcenter of an acuted angled triangle $ABC$ . $D$ and $E$ are feets of perpendiculars from $A$ and $B$ on $BC$ and $AC$ respectively. Let $OD$ and $OE$ intersect $BE$ and $AD$ in $K$ and $L$ , respectively. Let $X$ be intersection of circumcircles of $HKD$ and $HLE$ different than $H$ , and $M$ is midpoint of $AB$ . Prove that $K, L, M$ are collinear iff $X$ is circumcenter of $EOD$ .