A convex polygon does not extend outside a square side 1. Prove that the sum of the squares of its sides is at most 4.
Form a right-angled triangle on each side of the polygon (and outside it), by taking the other two sides parallel to the sides of the square. The sum of the squares of the polygon's sides equals the sum of the squares of the non-hypoteneuse sides of the triangles. Because the polygon is convex, these triangle sides form 4 sets, one for each side of the square, and each set having lengths totalling less than 1 (the side of the square). So the sum of the squares in each set is less than 1 (∑x2 < (∑x)2 = 1).
27th Putnam 1966
© John Scholes
25 Jan 2002