# Solution: 2016-23 Inequality on complex numbers

Suppose that $$z_1, z_2, \dots, z_n$$ are complex numbers satisfying $$\sum_{k=1}^n z_k = 0$$. Prove that
$\sum_{k=1}^n |z_{k+1} – z_k|^2 \geq 4 \sin^2 \left( \frac{\pi}{n} \right) \sum_{k=1}^n |z_k|^2,$
where we let $$z_{n+1} = z_1$$.

The best solution was submitted by Kim, Taegyun (김태균, 2016학번). Congratulations!

Here is his solution of problem 2016-23.

Alternative solutions were submitted by 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution), 국윤범 (수리과학과 2015학번, +3), 김기현 (수리과학과 대학원생, +3, alternative solution), 이상민 (수리과학과 2014학번, +2). One incorrect solution was submitted.

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# Solution: 2016-22 Computing the Determinant

Let $$M_n=(a_{ij})_{ij}$$ be an $$n\times n$$ matrix such that $a_{ij}=\binom{2(i+j-1)}{i+j-1}.$ What is $$\det M_n$$?

The best solution was submitted by Koon, Yun Bum (국윤범, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-22.

Alternative solutions were submitted by 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 이상민 (수리과학과 2014학번, +3), 채지석 (2016학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 홍진표 (서울대학교 재료공학부 2013학번, +3).

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Suppose that $$z_1, z_2, \dots, z_n$$ are complex numbers satisfying $$\sum_{k=1}^n z_k = 0$$. Prove that
$\sum_{k=1}^n |z_{k+1} – z_k|^2 \geq 4 \sin^2 \left( \frac{\pi}{n} \right) \sum_{k=1}^n |z_k|^2,$
where we let $$z_{n+1} = z_1$$.