Let \( A, B, C \) be \( N \times N \) Hermitian matrices with \( C = A+B \). Let \( \alpha_1 \geq \dots \geq \alpha_N \), \( \beta_1 \geq \dots \geq \beta_N \), \( \gamma_1 \geq \dots \geq \gamma_N \) be the eigenvalues of \( A, B, C \), respectively. For any \( 1 \leq k \leq N \), prove that
\[ \gamma_1 + \gamma_2 + \dots + \gamma_k \leq (\alpha_1 + \alpha_2 + \dots + \alpha_k) + (\beta_1 + \beta_2 + \dots + \beta_k) \]

The best solution was submitted by Sounggun Wee (위성군, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2017-01.

Alternative solutions were submitted by 강한필 (2016학번, +3), 김태균 (수리과학과 2016학번, +3), 배형진 (마포고 3학년, +3), 오동우 (수리과학과 2015학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이정환 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), Huy Tung Nguyen (2016학번, +3), 곽상훈 (수리과학과 2013학번, +3), 이본우 (2017학번, +3), 이태영 (수리과학과 2013학번, +2).

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.

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).

For a positive integer \( n \), let \( d(n) \) be the number of positive divisors of \( n \). Prove that, for any positive integer \( M \), there exists a constant \( C>0 \) such that \( d(n) \geq C ( \log n )^M \) for infinitely many \( n \).

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

Here is his solution of problem 2016-21.

Alternative solutions were submitted by 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution), 국윤범 (수리과학과 2015학번, +3), 이상민 (수리과학과 2014학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이정환 (수리과학과 2015학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3), 조준영 (수리과학과 2012학번, +3), 이시우 (포항공대 수학과 2013학번, +3).