A square $ABCD$ is inscribed in a circle. If $M$ is a point on the shorter arc $AB$, prove that \[MC \cdot MD > 3\sqrt{3} \cdot MA \cdot MB.\]

Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$.

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 $FDE$.