Puzzles

This is a list of small mathematical puzzles I've written which I find fun --- it's a work in progress though :) All of these problems were written by myself, for some of them I checked my answers and/or did some tedious details with AI.

Problem 1. Consider three numbers α , β , γ ∈ [ 0 , π ] such that for any x , y , z ∈ ℝ we have x 2 + y 2 + z 2 + 2 x y cos ⁡ α + 2 x z cos ⁡ β + 2 y z cos ⁡ γ ≥ 0 Then α + β + γ ≤ 2 π .

Hint:Cholesky decomposition of quadratic form.

Problem 2. In the same conditions as in Problem 1, prove that if α ≤ β ≤ γ then γ ≤ α + β .

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?

Problem 4. Consider a circle. Repeatedly draw chords on this circle uniformly at random until two intersect. What is the expected number of chords you draw?

Solution: e 2 − 1 2