Suppose that $\Pi$ is a closed polygon in the plane. If $\Pi$ is equilateral $k$-gon, and if $A$ is the area of $\Pi$, and $L$ the length of its boundary, prove that
$$\frac{A}{L^2} \leq \frac{1}{4k} \cot \frac{\pi}{k} \leq \frac{1}{4\pi}.$$

The best solution was submitted by 윤창기 (서울대학교 화학과). Congratulations!

Here is his solution of problem 2019-01.

Similar solutions were submitted by 길현준 (2018학번, +3), 조재형 (수리과학과 2016학번, +3), 김태균 (수리과학과 2016학번, +2). Alternative solution was submitted by 고성훈 (2018학번, +3). Four incorrect solutions were received.