KURSCHAK 2000

Paint the grid points of $L=\{0,1,\dots,n\}^2$ with red or green in such a way that every unit lattice square in $L$ has exactly two red vertices. How many such colorings are possible?

Let $ABC$ be a non-equilateral triangle in the plane, and let $T$ be a point different from its vertices. Define $A_T$ , $B_T$ and $C_T$ as the points where lines $AT$ , $BT$ , and $CT$ meet the circumcircle of $ABC$ . Prove that there are exactly two points $P$ and $Q$ in the plane for which the triangles $A_PB_PC_P$ and $A_QB_QC_Q$ are equilateral. Prove furthermore that line $PQ$ contains the circumcenter of $\triangle ABC$ .

Let $k\ge 0$ be an integer and suppose the integers $a_1,a_2,\dots,a_n$ give at least $2k$ different residues upon division by $(n+k)$ . Show that there are some $a_i$ whose sum is divisible by $n+k$ .