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Home » Olympiad Resources » International Competitions » Balkan MO » 2008

Balkan MO 2008

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06 May 2008

1	Given a scalene acute triangle $ABC$ with $AC>BC$ let $F$ be the foot of the altitude from $C$ . Let $P$ be a point on $AB$ , different from $A$ so that $AF=PF$ . Let $H,O,M$ be the orthocenter, circumcenter and midpoint of $[AC]$ . Let $X$ be the intersection point of $BC$ and $HP$ . Let $Y$ be the intersection point of $OM$ and $FX$ and let $OF$ intersect $AC$ at $Z$ . Prove that $F,M,Y,Z$ are concyclic.	S
2	Is there a sequence $a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $n$ : a) $a_1+a_2+\ldots+a_n\le n^2$ b) $\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}\le2008$ ?	S
3	Let $n$ be a positive integer. Consider a rectangle $(90n+1)\times(90n+5)$ consisting of unit squares. Let $S$ be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of $S$ is divisible by $4$ .	S
4	Let $c$ be a positive integer. The sequence $a_1,a_2,\ldots$ is defined as follows $a_1=c$ , $a_{n+1}=a_n^2+a_n+c^3$ for all positive integers $n$ . Find all $c$ so that there are integers $k\ge1$ and $m\ge2$ so that $a_k^2+c^3$ is the $m$ th power of some integer.	S

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