KURSCHAK 1988

Prove that if there exists a point $P$ inside the convex quadrilateral $ABCD$ such that the triangles $PAB$ , $PBC$ , $PCD$ , $PDA$ have the same area, then one of the diagonals of $ABCD$ bisects the area of the quadrilateral.

Set $T\subset\{1,2,\dots,n\}^3$ has the property that for any two triplets $(a,b,c)$ and $(x,y,z)$ in $T$ , we have $a<b<c$ , and also, we know that at most one of the equalities $a=x$ , $b=y$ , $c=z$ holds. Maximize $|T|$ .

Consider the convex lattice quadrilateral $PQRS$ whose diagonals intersect at $E$ . Prove that if $\angle P+\angle Q<180^\circ$ , then the $\triangle PQE$ contains inside it or on one of its sides a lattice point other than $P$ and $Q$ .