A hungry lion runs inside a circus arena which is a circle of radius 10 meters. Running in broken lines (i.e. along a piecewise linear trajectory), the lion covers 30 kilometers. Prove that the sum of all turning angles is at least 2998 radians.
Source: The Puzzle Toad
Imagine the lion is static and instead the center of the arena moves around. Then, the problem translates to a point inside the arena alternating between traveling straight South and then moving north along arcs around the center of the arena. Since the total distance traveled straight South is 30KM and the distance between the starting and the ending points is at most 20M, the total distance traveled North must be at least 30KM – 20M = 29980M. The radius of the arena is 10M, so this corresponds to at least 2998 radians.