KURSCHAK 2001

Let $k\ge 3$ be an integer. Prove that if $n>\binom k3$ , then for any $3n$ pairwise different real numbers $a_i,b_i,c_i$ ( $1\le i\le n$ ), among the numbers $a_i+b_i$ , $a_i+c_i$ , $b_i+c_i$ , one can find at least $k+1$ pairwise different numbers. Show that this is not always the case when $n=\binom k3$ .

In a square lattice let us take a lattice triangle that has the smallest area among all the lattice triangles similar to it. Prove that the circumcenter of this triangle is not a lattice point.