IMO 1976 SL 3

Origin: CZS

Problem

In a convex quadrangle with area 32 cm2, the sum of the lengths of two nonadjacent edges and of the length of one diagonal is equal to 16 cm. (a) What is the length of the other diagonal? (b) What are the lengths of the edges of the quadrangle if the perimeter is a minimum? (c) Is it possible to choose the edges in such a way that the perimeter is a maximum?

Solution

(a) Let ABCD be a quadrangle with 16 = d = AB + CD + AC, and let S be its area. Then S \leq(AC \cdot AB + AC \cdot CD)/2 = AC(d −AC)/2 \leq d2/8 = 32, where equality occurs if and only if AB \perpAC \perpCD and AC = AB + CD = 8. In this case BD = 8 \sqrt 2. (b) Let A′ be the point with −−\to DA′ = −\to AC. The triangular inequality implies AD + BC \geqAA′ = 8 \sqrt