For \(n\ge 1\), let \(f(x)=x^n+\sum_{k=0}^{n-1} a_k x^k \) be a polynomial with real coefficients. Prove that if \(f(x)>0\) for all \(x\in [-2,2]\), then \(f(x)\ge 4\) for some \(x\in [-2,2]\).

The best solution was submitted by Choi, Inhyeok (최인혁, 물리학과 2015학번). Congratulations!

Here is his solution of problem 2018-02.

Alternative solutions were submitted by 이본우 (수리과학과 2017학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이종원 (수리과학과 2014학번, +3), 채지석 (수리과학과 2016학번, +3), 하석민 (수리과학과 2017학번, +3), 한준호 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +2), 이재우 (함양고등학교 3학년, +2).

For an integer \( n \geq 4 \), find the solutions of the equation
\[
\sum_{k=1}^n \frac{\sin \frac{k\pi}{n+1}}{\sin (\frac{k\pi}{n+1} -x)} = 0.
\]

The best solution was submitted by Choi, Inhyeok (최인혁, 물리학과 2015학번). Congratulations!

Here is his solution of problem 2017-03.

Alternative solutions were submitted by 위성군 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), Huy Tung Nguyen (2016학번, +3), 조정휘 (건국대학교 수학과 2014학번, +3), 배형진 (마포고 3학년, +2), 오동우 (수리과학과 2015학번, +2).

Let \(k\) be a positive integer. Let \(a_n=1\) if \(n\) is not a multiple of \(k+1\), and \(a_n=-k\) if \(n\) is a multiple of \(k+1\). Compute \[\sum_{n=1}^\infty \frac{a_n}{n}.\]

The best solution was submitted by Choi, Inhyeok (최인혁, 물리학과 2015학번). Congratulations!

Here is his solution of problem 2016-12.

Alternative solutions were submitted by 강한필 (2016학번, +3), 국윤범 (수리과학과 2015학번, +3), 김기택 (수리과학과 2015학번, +3), 김재현 (2016학번, +3), 김태균 (2016학번, +3), 김태형 (EEWS대학원 석사과정, +3), 박찬우 (서울대학교 통계학과 2016학번, +3), 신준형 (수리과학과 2015학번, +3), 오동우 (2015학번, +3), 유찬진 (수리과학과 2015학번, +3), 이정환 (수리과학과 2015학번, +3), 이종원 (수리과학과 2014학번, +3), 임성혁 (2016학번, +3), 장기정 (수리과학과 2014학번, +3), 채지석 (2016학번, +3), 최대범 (2016학번, +3), 한준호 (수리과학과 2015학번, +3), Muhammaadfiruz Hasanov (2014학번, +3), 박기연 (2016학번, +2), 박진호 (물리학과 2015학번, +2), 송교범 (서대전고등학교 3학년, +2), 송민학 (월촌중학교 3학년, +2), 윤준기 (전기및전자공학부 2014학번, +2), 정성진 (수리과학과 2013학번, +2), 이본우 (대구과학고등학교 3학년, +2), 이상민 (수리과학과 2014학번, +2), 이시우 (포항공대 수학과 2013학번, +2). There were 2 incorrect solutions and 2 submissions by email missing attachments.

Find all positive integers \(n\) such that the following statement holds:

Let \(f:\mathbb{R}^n\to \mathbb {R}\) be a differentiable function that has a unique critical point \(c\). If \(f\) has a local maximum at \(c\), then \(f(c)\) is an absolute maximum of \(f\).

The best solution was submitted by Choi, Inhyeok (최인혁, 2015학번). Congratulations!

Here is his solution of problem 2015-14.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 김재준 (2014학번, +3), 박훈민 (수리과학과 2013학번, +3), 박성혁 (수리과학과 2014학번, +3), 오동우 (2015학번, +3), 이영민 (수리과학과 2012학번, +3), 장기정 (수리과학과 2014학번, +3), 신준형 (2015학번, +2). One incorrect solutions were received (LAL). Delayed submissions were not graded.