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 $\gamma \le \alpha +\beta$.

Problem 3. Alice and Bob play tag on a circle. Each of them starts at some point, and their starting points are distinct. After an eventful game, they return to their respective starting positions. Alice notices that she went around the circle in a clockwise direction 2025 times, whereas Bob did 2026 times. Has Alice necessarily caught Bob at least once?