A polygon is $\textit{monochromatic}$ if all its vertices are coloured by the same colour. Suppose now every point of the plane is coloured red or blue. Show that there exists either a monochromatic equilateral triangle of side length $2$, or a monochromatic equilateral triangle of side length $\sqrt{3}$, or a monochromatic rhombus of side length $1$.

Let $a,b,c$ be distinct nonzero real numbers. If the equations $ax^3+bx+c=0$, $bx^3+cx+a=0,$ and $cx^3+ax+b=0$ have a common root, prove that at least one of these equations has three real roots(not necessarily distinct).

Find all pairs $(a,b)$ of integers $a$ and $b$ satisfying \[(b^2+11(a-b))^2=a^3 b\]

Let $\triangle ABC$ be a scalene triangle, and let $D$ and $E$ be points on sides $AB$ and $AC$ respectively such that the circumcircles of triangles $\triangle ACD$ and $\triangle ABE$ are tangent to $BC$. Let $F$ be the intersection point of $BC$ and $DE$. Prove that $AF$ is perpendicular to the Euler line of $\triangle ABC$.