Puzzles

This is a list of small mathematical puzzles I've written which I find fun --- it's a work in progress though :)

Problem 1. Consider three numbers $\alpha ,\beta ,\gamma \in [0,\pi ]$ such that for any $x,y,z\in ℝ$ we have $$x^2+y^2+z^2+2xy\cos \alpha +2xz\cos \beta +2yz\cos \gamma \ge 0$$ Then $\alpha +\beta +\gamma \le 2\pi$.

Problem 2. In the same conditions as in Problem 1, prove that if $\alpha \le \beta \le \gamma$ then $\gamma \le \alpha +\beta$.