Problem

Source: Ukraine 2005 grade 10

Tags: trigonometry



Let $ 0 \le \alpha, \beta, \gamma \le \frac{\pi}{2}$ satisfy the conditions $ \sin \alpha+\sin \beta+\sin \gamma=1, \sin \alpha \cos 2\alpha+\sin \beta\cos 2\beta+\sin \gamma \cos 2\gamma=-1.$ Find all possible values of $ \sin^2 \alpha+\sin^2 \beta+\sin^2 \gamma$.