Let $ABC$ be a triangle with $\angle BAC\ne90^\circ$. Let $O$ be its circumcenter and $\Gamma$ the circumcircle of triangle $BOC$. Suppose that $\Gamma$ intersects segment $AB$ at $P\ne B$, and segment $AC$ at $Q\ne C$. Let $ON$ be a diameter of $\Gamma$. Prove that quadrilateral $APNQ$ is a parallelogram.
For a positive integer $k$, call an integer a pure $k$-th power if it can be represented as $m^k$ for some integer $m$. Show that for every positive integer $n$ there exist $n$ distinct positive integers whose sum is a pure 2009-th power and whose product is a pure 2010-th power.
Let $n$ be a positive integer. At a party of $n$ people, each pair either are acquainted with each other or are not. What is the maximum possible number of pairs who are not acquainted but have a common acquaintance among the participants?
Let $ABC$ be an acute triangle with $AB\gt BC$ and $AC\gt BC$. Let $O,H$ be its circumcenter and orthocenter. The circumcircle of $AHC$ intersects line $AB$ again at $M\ne A$, and the circumcircle of $AHB$ intersects line $AC$ again at $N\ne A$. Prove that the circumcenter of triangle $MNH$ lies on line $OH$.
