JBMO 2001

Solve the equation $a^3+b^3+c^3=2001$ in positive integers.

Mircea Becheanu, Romania

Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $CA \neq CB$ . Let $CH$ be an altitude and $CL$ be an interior angle bisector. Show that for $X \neq C$ on the line $CL$ , we have $\angle XAC \neq \angle XBC$ . Also show that for $Y \neq C$ on the line $CH$ we have $\angle YAC \neq \angle YBC$ .

Bulgaria

Let $ABC$ be an equilateral triangle and $D$ , $E$ points on the sides $[AB]$ and $[AC]$ respectively. If $DF$ , $EF$ (with $F\in AE$ , $G\in AD$ ) are the interior angle bisectors of the angles of the triangle $ADE$ , prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$ . When does the equality hold?

Greece

Let $N$ be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of $N$ which form a triangle of area smaller than 1.