Find the minimum value of
\[
\int_{\mathbb{R}} f(x) \log f(x) dx
\]
among functions \(f: \mathbb{R} \rightarrow \mathbb{R}^+ \cup \{ 0 \}\) that satisfy the condition
\[
\int_{\mathbb{R}} f(x) dx = \int_{\mathbb{R}} x^2 f(x) dx = 1.
\]

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-13.

Alternative solutions were submitted by 김경석 (2015학번, +3), 김재준 (2014학번, +3), 김희주 (2015학번, +2), 박성혁 (수리과학과 2014학번, +2), 박훈민 (수리과학과 2013학번, +3), 신준형 (2015학번, +3), 오동우 (2015학번, +2), 이신영 (물리학과 2012학번, +2), 이영민 (수리과학과 2012학번, +2), 이정환 (2015학번, +3), 장기정 (수리과학과 2014학번, +2), 최인혁 (2015학번, +2), Luis F. Abanto-Leon (+2), 이시우 (포항공대 수학과 2013학번, +3). Two incorrect solutions (L.S.M., H.I.S.) were submitted.

Let \(A\) be an \(n\times n\) matrix with complex entries. Prove that if \(A^2=A^*\), then \[\operatorname{rank}(A+A^*)=\operatorname{rank}(A).\] (Here, \(A^*\) is the conjugate transpose of \(A\).)

The best solution was submitted by Kim, Kee Taek (김기택, 2015학번). Congratulations!

Here is his solution of problem 2015-12.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 박지민 (전산학과 대학원생, +3), 엄태현 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3).

Does \(\frac{1}{n \sin n}\) converge as \(n\) goes to infinity?

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-11.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 신준형 (2015학번, +3), 엄태현 (수리과학과 2012학번, +3), 오동우 (2015학번, +3), 이수철 (수리과학과 2012학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3), 이상민 (수리과학과 2014학번, +2), 이영민 (수리과학과 2012학번, +2). One incorrect solution (KDR) was submitted.

Let \(w_1,w_2,\ldots,w_n\) be positive real numbers such that \( \sum_{i=1}^n w_i=1\). Prove that if \(x_1,x_2,\ldots,x_n\in [0,\pi]\), then \[ \sin \left(\prod_{i=1}^n x_i^{w_i} \right) \ge \prod_{i=1}^n (\sin x_i)^{w_i}.\]

The best solution was submitted by Lee, Young Min (이영민, 수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-10.

Other (but mostly identical) solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 엄태현 (수리과학과 2012학번, +3), 오동우 (2015학번, +3), 이수철 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3).

Let \(n\ge 1\) and \(a_0,a_1,a_2,\ldots,a_{n}\) be non-negative integers. Prove that if \[ N=\frac{a_0^2+a_1^2+a_2^2+\cdots+a_{n}^2}{1+a_0a_1a_2\cdots a_{n}}\] is an integer, then \(N\) is the sum of \(n\) squares of integers.

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-9.

Alternative solutions were submitted by 김기현 (수리과학과 2012학번, +3), 엄태현 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3), 윤지훈 (2012학번, +2). One incorrect solution was submitted (YSC).

Does there exist a subset \(A\) of \(\mathbb{R}^2\) such that \( \lvert A\cap L\rvert=2\) for every straight line \(L\)?

The best solution was submitted by Lee, Su Cheol (이수철, 수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-08.

Alternative solutions were submitted by 김기현 (수리과학과 2012학번, +3), 김동률 (2015학번, +3), 엄태현 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 진우영 (수리과학과 2012학번, +3), 박훈민 (수리과학과 2013학번, +2), 오동우 (2015학번, +2).

Prove or disprove that \[ \sum_{i=0}^r (-1)^i \binom{i+k}{k} \binom{n}{r-i} = \binom{n-k-1}{r}\] if \(k, r\) are non-negative integers and \(0\le r\le n-k-1\).

The best solution was submitted by Chin, Wooyoung (진우영, 수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-7.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 박훈민 (수리과학과 2013학번, +3), 엄태현 (수리과학과 2012학번, +3), 오동우 (2015학번, +3), 윤준기 (수리과학과 2014학번, +3), 이수철 (수리과학과 2012학번, +3), 이영민 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 최인혁 (2015학번, +3), 함도규 (2015학번, +3), 김성민 (캠브리지대학 진학 예정, +3).

Let \(A\) be an unbounded subset of the set \(\mathbb R\) of the real numbers. Let \(T\) be the set of all real numbers \(t\) such that \(\{tx-\lfloor tx\rfloor : x\in A\}\) is dense in \([0,1]\). Is \(T\) dense in \(\mathbb R\)?

The best solution was submitted by Kim, Kihyun (김기현, 수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-6.

Alternative solutions were submitted by 엄태현 (수리과학과 2012학번, +3), 이명재 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 최인혁 (2015학번, +3), 진우영 (수리과학과 2012학번, +3), 배형진 (마포고 1학년, +2). One incorrect solution was submitted (KDR).

Determine all \(n\times n\) matrices A such that \( \operatorname{tr}(AXY)=\operatorname{tr}(AYX)\) for all \(n\times n\) matrices \(X\) and \(Y\).

The best solution was submitted by Choi, Doo Seong (최두성, 수리과학과 2011학번). Congratulations!

Here is his solution of problem 2015-5.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 김경석 (2015학번, +3), 엄태현 (수리과학과 2012학번, +3), 오동우 (2015학번, +3), 유찬진 (2015학번, +3), 이수철 (수리과학과 2012학번, +3), 이명재 (수리과학과 2012학번, +3), 이영민 (수리과학과 2012학번, +3), 장기정 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 진우영 (수리과학과 2012학번, +3), 최인혁 (2015학번, +3), 함도규 (2015학번, +3), 홍혁표 (수리과학과 2013학번, +3), 박성혁 (수리과학과 2014학번, +2), 이종원 (수리과학과 2014학번, +2), 전한솔 (고려대, +3), 어수강 (서울대 수리과학부 대학원생, +3).

Let \( M=\begin{pmatrix} A & B \\ B^*& C \end{pmatrix}\) be a positive semidefinite Hermian matrix. Prove that \[ \operatorname{rank} M \le \operatorname{rank} A +\operatorname{rank} C.\] (Here, \(A\), \(B\), \(C\) are matrices.)

The best solution was submitted by 엄태현 (수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-04.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김경석 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 박성혁 (수리과학과 2014학번, +3), 오동우 (2015학번, +3), 이명재 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3, solution), 이종원 (수리과학과 2014학번, +3, solution), 장기정 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 진우영 (수리과학과 2012학번, +3), 최인혁 (2015학번, +3).