CMOQR 2021

Determine all real polynomials $p$ such that $p(x+p(x))=x^2p(x)$ for all $x$ .

Determine all integer solutions to the system of equations: \begin{align*} xy + yz + zx &= -4 x^2 + y^2 + z^2 &= 24 x^{3} + y^3 + z^3 + 3xyz &= 16 \end{align*}

$ABCDE$ is a regular pentagon. Two circles $C_1$ and $C_2$ are drawn through $B$ with centers $A$ and $C$ respectively. Let the other intersection of $C_1$ and $C_2$ be $P$ . The circle with center $P$ which passes through $E$ and $D$ intersects $C_2$ at $X$ and $AE$ at $Y$ . Prove that $AX = AY$ .

Let $O$ be the centre of the circumcircle of triangle $ABC$ and let $I$ be the centre of the incircle of triangle $ABC$ . A line passing through the point $I$ is perpendicular to the line $IO$ and passes through the incircle at points $P$ and $Q$ . Prove that the diameter of the circumcircle is equal to the perimeter of triangle $OPQ$ .