
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.
Puzzle from the archives.

The solution is provided HERE. Some notes regarding the 2998 radians bound…
As pointed out by one of the comments under the problem, while 2998 radians is a lower bound for the total turning, it may not be a tight bound. If the lion travels along the circumference of the arena, his total turning will be 3000 radians. He can lower the number to 2996 + π ~ 2999.14 radians by using the following strategy:
The lion travels 20M along the diagonal, then travel 29960M along the circumference, and then travels another 20M along the diagonal.
It is an interesting question to find a better strategy (if it exists) or to improve the proof and raise the lower bound.
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