Let
![\alpha,\beta,\gamma](/media/m/a/f/b/afb9db7af4c8576464589014659000d1.png)
be positive real numbers such that
![\alpha+\beta+\gamma < \pi](/media/m/1/c/9/1c9fbec9090d4a8799f0d228782d8ae8.png)
,
![\alpha+\beta > \gamma](/media/m/9/0/3/9033af970080688346cee5bbf07d3084.png)
,
![\beta+\gamma > \alpha](/media/m/6/0/e/60ea3226348a2246631decde8e909d01.png)
,
![\gamma + \alpha > \beta.](/media/m/f/1/1/f11a3a1dfb10ba42156d9f4fb2cbc4ec.png)
Prove that with the segments of lengths
![\sin \alpha, \sin \beta, \sin \gamma](/media/m/2/4/d/24d53beefc808ad6ae1293435ced1dfe.png)
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).](/media/m/7/2/2/722d6341f20bddcdd432285dbc7bf3e5.png)
Proposed by Soviet Union
%V0
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