BALKAN 1999

Let $O$ be the circumcenter of the triangle $ABC$ . The segment $XY$ is the diameter of the circumcircle perpendicular to $BC$ and it meets $BC$ at $M$ . The point $X$ is closer to $M$ than $Y$ and $Z$ is the point on $MY$ such that $MZ = MX$ . The point $W$ is the midpoint of $AZ$ .

a) Show that $W$ lies on the circle through the midpoints of the sides of $ABC$ ;

b) Show that $MW$ is perpendicular to $AY$ .

Let $p$ be an odd prime congruent to 2 modulo 3. Prove that at most $p-1$ members of the set $\{m^2 - n^3 - 1 \mid 0 < m,\ n < p\}$ are divisible by $p$ .

Let $ABC$ be an acute-angled triangle of area 1. Show that the triangle whose vertices are the feet of the perpendiculars from the centroid $G$ to $AB$ , $BC$ , $CA$ has area between $\frac 4{27}$ and $\frac 14$ .

Let $\{a_n\}_{n\geq 0}$ be a non-decreasing, unbounded sequence of non-negative integers with $a_0=0$ . Let the number of members of the sequence not exceeding $n$ be $b_n$ . Prove that $$(a_0 + a_1 + \cdots + a_m)( b_0 + b_1 + \cdots + b_n ) \geq (m + 1)(n + 1).$$