VERY HARD South Korean Geometry Problem (CSAT Exam)

Thanks to Hyeong-jun (H. J.) for emailing me this problem! This is a challenging problem from the math section of the 1997 CSAT, a standardized test in South Korea. Can you figure it out? It took me several attempts, but it was really satisfying when I solved it (I did need to look up one key insight, which is the first thing presented in the solution). So give it a try, and don't peek at the rest of the solution too early!

0:00 Problem
1:58 My many attempts
3:28 Unwrapping a cone
5:14 Solution

Coordinate geometry solution and proof of uphill/downhill track
https://www.desmos.com/calculator/l18...

Note: pretty much every time I said or wrote "circular arc" I meant to say or write "circular sector." A "circular arc" is a portion of a circle's circumference; a "circular sector" is the enclosed region between the arc and two radii.

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