RS-SMO 2010

In an acute-angled triangle $ABC$ , $M$ is the midpoint of side $BC$ , and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$ , respectively. Let $H$ be the orthocenter of $\Delta ABC$ , $S$ the midpoint of $AH$ , and $G$ the intersection of $FE$ and $AH$ . If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$ , prove that $\angle HMA = \angle GNS$ .

Proposed by Marko Djikic

Let $O$ be the circumcenter of triangle $ABC$ . A line through $O$ intersects the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of $ABO$ again at point $P \neq O$ inside the triangle. A point $Q$ on side $AB$ is such that $\frac{AQ}{QB}=\frac{DP}{PE}$ . Prove that $\angle APQ = 2\angle CAP$ .

Proposed by Dusan Djukic

Let $a_0$ and $a_n$ be different divisors of a natural number $m$ , and $a_0, a_1, \ldots, a_n$ be a sequence of natural numbers such that it satisfies $$a_{i+1} = |a_i\pm a_{i-1}|\text{ for }0 < i < n$$ If $gcd(a_0,a_1,\ldots, a_n) = 1$ , show that there exists a term of the sequence that is smaller than $\sqrt{m}$ .

Proposed by Dusan Djukic