CMOQR 2018

Determine all real solutions to the following system of equations: $$\begin{cases} y = 4x^3 + 12x^2 + 12x + 3 x = 4y^3 + 12y^2 + 12y + 3. \end{cases}$$

We call a pair of polygons, $p$ and $q$ , nesting if we can draw one inside the other, possibly after rotation and/or reflection; otherwise we call them non-nesting.

Let $p$ and $q$ be polygons. Prove that if we can find a polygon $r$ , which is similar to $q$ , such that $r$ and $p$ are non-nesting if and only if $p$ and $q$ are not similar.

Let $ABC$ be a triangle with $AB = BC$ . Prove that $\triangle ABC$ is an obtuse triangle if and only if the equation $$Ax^2 + Bx + C = 0$$ has two distinct real roots, where $A$ , $B$ , $C$ , are the angles in radians.

Construct a convex polygon such that each of its sides has the same length as one of its diagonals and each diagonal has the same length as one of its sides, or prove that such a polygon does not exist.