RMM 2008

Let $ABC$ be an equilateral triangle and $P$ in its interior. The distances from $P$ to the triangle's sides are denoted by $a^2, b^2,c^2$ respectively, where $a,b,c>0$ . Find the locus of the points $P$ for which $a,b,c$ can be the sides of a non-degenerate triangle.

Prove that every bijective function $f: \mathbb{Z}\rightarrow\mathbb{Z}$ can be written in the way $f=u+v$ where $u,v: \mathbb{Z}\rightarrow\mathbb{Z}$ are bijective functions.

Let $a>1$ be a positive integer. Prove that every non-zero positive integer $N$ has a multiple in the sequence $(a_n)_{n\ge1}$ , $a_n=\left\lfloor\frac{a^n}n\right\rfloor$ .

Consider a square of sidelength $n$ and $(n+1)^2$ interior points. Prove that we can choose $3$ of these points so that they determine a triangle (eventually degenerated) of area at most $\frac12$ .