23rd All Russian Mathematical Olympiad Problems 1997

1.  p(x) is a quadratic polynomial with non-negative coefficients. Show that p(xy)2 ≤ p(x2)p(y2). 2.  A convex polygon is invariant under a 90o rotation. Show that for some R there is a circle radius R contained in the polygon and a circle radius R√2 which contains the polygon. 3.  A rectangular box has integral sides a, b, c, with c odd. Its surface is