KURSCHAK 1997

Let $p>2$ be a prime number and let $L=\{0,1,\dots,p-1\}^2$ . Prove that we can find $p$ points in $L$ with no three of them collinear.

The center of the circumcircle of $\triangle ABC$ is $O$ . The incenter of the triangle is $I$ , and the intouch triangle is $A_1B_1C_1$ . Let $H_1$ be the orthocenter of $\triangle A_1B_1C_1$ . Prove that $O$ , $I$ , and $H_1$ are collinear.

Prove that the vertices of any planar graph can be colored with $3$ colors such that there is no monochromatic cycle.