ILL 1987 Problem 68

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 $$A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).$$

Proposed by Soviet Union