IMO 1987 SL 19

Origin: USS

Problem

Let \alpha, \beta, \gamma be positive real numbers such that \alpha + \beta + \gamma < \pi, \alpha + \beta > \gamma, \beta + \gamma > \alpha, \gamma + \alpha > \beta. Prove that with the segments of lengths sin \alpha, sin \beta, sin \gamma we can construct a triangle and that its area is not greater than 8(sin 2\alpha + sin 2\beta + sin 2\gamma).

Solution

The facts given in the problem allow us to draw a triangular pyramid with angles 2\alpha, 2\beta, 2\gamma at the top and lateral edges of length 1/2. At the base there is a triangle whose side lengths are exactly sin \alpha, sin \beta, sin \gamma. The area of this triangle does not exceed the sum of areas of the lateral sides, which equals (sin 2\alpha + sin 2\beta + sin 2\gamma)/8.