1 - st Mediterranean Mathematical Competition 1998

1. A square $ABCD$ is inscribed in a circle. If $M$ is a point on the shorter arc $AB$, prove that 

$MC· MD > 3\sqrt{3} · MA· MB$. 

(Greece) 

2. Prove that the polynomial 

$z^{2n} + z^ n + 1$ (n ∈ N) 

is divisible by the polynomial $z^ 2 +z+1$ if and only if n is not a multiple of $3$. 

(Croatia) 

3. In a triangle $ABC$, $I$ is the incenter and $D,E,F$ are the points of tangency of the incircle with $BC,CA,AB$, respectively. The bisector of angle $BIC$ meets $BC$ at $M$, and the line $AM$ intersects $EF$ at $P$. Prove that $DP$ bisects the angle $FDE$. 

(Spain)