## IMO 2009 Germany – Day 1 Problems ខែកក្កដា 16, 2009

Posted by តារារស្មី in គណិតវិទ្យា.

Problem 1. Let $n$ be a positive integer and let $a_1,a_2,a_3,...,a_k$ ( $k\geq 2$) be distinct integer in the set { 1,2,…,n} such that $n$ divides $a_i(a_{i+1}-1)$ for $i = 1,2,...,k-1$. Prove that $n$ does not divide $a_k(a_1-1)$.

Problem 2. Let ABC be a triangle with circumcenter O. The points P and Q are interior points of the sides CA and AB respectively. Let K,L and M be the midpoints of the segments BP,CQ and PQ. respectively, and let Γ be the circle passing through K,L and M. Suppose that the line PQ is tangent to the circle Γ. Prove that OP = OQ.

Problem 3. Suppose that $s_1,s_2,s_3, ...$ is a strictly increasing sequence of positive integers such that the sub-sequences ss_{1},ss_{2},ss_{3}, … and ss_{1+1},ss_{2+1},ss_{3+1}, … are both arithmetic progressions. Prove that the sequence s1,s2,s3, … is itself an arithmetic progression.