VWO 1998

Prove there exist positive integers a,b,c for which $a+b+c=1998$ , the gcd is maximized, and $0<a<b\leq c<2a$ . Find those numbers. Are they unique?

Given a cube with edges of length 1, $e$ the midpoint of $[bc]$ , and $m$ midpoint of the face $cdc_1d_1$ , as on the figure. Find the area of intersection of the cube with the plane through the points $a,m,e$ .

A billiard table. (see picture) A white ball is on $p_1$ and a red ball is on $p_2$ . The white ball is shot towards the red ball as shown on the pic, hitting 3 sides first. Find the minimal distance the ball must travel.