Does there exist an infinite sequence such that (i) every integer appears infinitely many times and (ii) the sequence is periodic modulo \(m\) for every positive integer \(m\)?

The best solution was submitted by Park, Sunghyuk (박성혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-15.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 신준형 (2015학번, +3), 최인혁 (2015학번, +2), 이영민 (수리과학과 2012학번, +2), 장기정 (수리과학과 2014학번, +2).

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.

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).