Let \(m\) and \(n\) be odd integers. Determine \[ \sum_{k=1}^\infty \frac{1}{k^2}\tan\frac{k\pi}{m}\tan \frac{k\pi}{n}.\]

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Let \(m\) and \(n\) be odd integers. Determine \[ \sum_{k=1}^\infty \frac{1}{k^2}\tan\frac{k\pi}{m}\tan \frac{k\pi}{n}.\]

Prove that if a finite ring has two elements \(x\) and \(y\) such that \(xy^2=y\), then \( yxy=y\).

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is Solution of Problem 2012-16.

Alternative solutions were submitted by 김주완 (수리과학과 2010학번, +3), 김지원 (수리과학과 2010학번, +3), 서기원 (수리과학과 2009학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 박민재 (2011학번, +3), 조상흠 (수리과학과 2010학번, +3), 정우석 (서강대 수학과 2011학번, +3). One incorrect solution (KHK) was submitted.

Prove that if a finite ring has two elements \(x\) and \(y\) such that \(xy^2=y\), then \( yxy=y\).

Let \(n\) be a fixed positive integer. Find all functions \( f:\mathbb{R}\to\mathbb{R}\) satisfying \[ f(x^{n+1}-y^{n+1})=(x-y)[f(x)^n+f(x)^{n-1}f(y)+\cdots+f(x)f(y)^{n-1}+f(y)^n].\]

The best solution was submitted by Kim, Taeho (김태호), 수리과학과 2011학번. Congratulations!

Here is his Solution of Problem 2012-15.

Alternative solutions were submitted by 임정환 (수리과학과 2009학번, +3), 곽걸담 (물리학과 2011학번, +2), 서기원 (수리과학과 2009학번, +2), 김홍규 (수리과학과 2011학번, +2), 김지원 (수리과학과 2010학번, +2), 이명재 (2012학번, +2), 조상흠 (수리과학과 2010학번, +2). There were 2 incorrect submissions (LHJ, KDR).

Let \(n\) be a fixed positive integer. Find all functions \( f:\mathbb{R}\to\mathbb{R}\) satisfying \[ f(x^{n+1}-y^{n+1})=(x-y)[f(x)^n+f(x)^{n-1}f(y)+\cdots+f(x)f(y)^{n-1}+f(y)^n].\]

Determine all continuous functions \(f:(0,\infty)\to(0,\infty)\) such that \[ \int_t^{t^3} f(x) \, dx = 2\int_1^t f(x)\,dx\] for all \(t>0\).

The best solution was submitted by Junghwan Lim (임정환), 수리과학과 2009학번. Congratulations!

Here is his Solution of Problem 2012-14.

Alternative solutions were submitted by 김주완 (2010학번, +3), 김태호 (수리과학과 2011학번, +3), 김홍규 (2011학번, 3), 곽걸담 (물리학과 2011학번, +3), 이신영 (2012학번, +3), 박민재 (2011학번, +3), 박종호 (수리과학과 2009학번, +3), 서기원 (수리과학과 2009학번, +3), 윤영수 (2011학번, +3), 이명재 (2012학번, +3), 조상흠 (2010학번, +3), 조준영 (2012학번, +3), 양지훈 (수리과학과 2010학번, +2), 최원준 (물리학과 2009학번, +2), 장영재 (수리과학과 2011학번, +2), 김건수 (서울대학교 전기컴퓨터공학부 2012학번, +3), 고재윤 (연세대학교, +3), 박훈민 (대전과학고 3학년, +3), 박항 (한국과학영재학교 2010학번, +3), 어수강 (서울대학교 수리과학부 대학원생, +3). There were 3 incorrect solutions submitted (RJH, KDR, JWS).

Determine all continuous functions \(f:(0,\infty)\to(0,\infty)\) such that \[ \int_t^{t^3} f(x) \, dx = 2\int_1^t f(x)\,dx\] for all \(t>0\).