KURSCHAK 2002

We have an acute-angled triangle which is not isosceles. We denote the orthocenter, the circumcenter and the incenter of it by $H$ , $O$ , $I$ respectively. Prove that if a vertex of the triangle lies on the circle $HOI$ , then there must be another vertex on this circle as well.

The Fibonacci sequence is defined as $f_1=f_2=1$ , $f_{n+2}=f_{n+1}+f_n$ ( $n\in\mathbb{N}$ ). Suppose that $a$ and $b$ are positive integers such that $\frac ab$ lies between the two fractions $\frac{f_n}{f_{n-1}}$ and $\frac{f_{n+1}}{f_{n}}$ . Show that $b\ge f_{n+1}$ .

Prove that the edges of a complete graph with $3^n$ vertices can be partitioned into disjoint cycles of length $3$ .