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