RO-NMO 2004 Problem 74

Let $\mathcal U = \left\{ \left( x,y \right) | x,y \in \mathbb Z, \ 0 \leq x,y < 4 \right\}$ .

(a) Prove that we can choose $6$ points from $\mathcal U$ such that there are no isosceles triangles with the vertices among the chosen points.

(b) Prove that no matter how we choose $7$ points from $\mathcal U$ , there are always three which form an isosceles triangle.

Radu Gologan, Dinu Serbanescu