Evaluate the following integral for \( z \in \mathbb{C}^+ \).\[\frac{1}{2\pi} \int_{-2}^2 \log (z-x) \sqrt{4-x^2} dx.\]

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

Here is his solution of problem 2015-16.

Alternative solutions were submitted by 최인혁 (2015학번, +2), 박훈민 (수리과학과 2013학번, +2), 박성혁/이경훈 (수리과학과 2014학번, +2).

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.

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 \(n\ge 1\) 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).

Let \( A\) be a set of \(n\ge 2\) odd integers. Prove that there exist two distinct subsets \(X\), \(Y\) of \(A\) such that \[ \sum_{x\in X} x\equiv\sum_{y\in Y}y \pmod{2^n}.\]

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

Here is his solution of problem 2015-1.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김경석 (2015학번, +3), 김기현 (2012학번, +3), 김동철 (2013학번, +3), 배형진 (마포고 1학년, +2), 어수강 (서울대 수리과학부 대학원생, +3), 엄태현 (2012학번, +3), 오동우 (2015학번, +3), 유찬진 (2015학번, +3), 윤성철 (홍익대 수학교육과, +3), 이명재 (수리과학과 2012학번, +3), 이병학 (2013학번, +3), 이상민 (수리과학과 2014학번, +3), 이수철 (2012학번, +3), 이시우 (POSTECH 수학과 2013학번, +3), 이영민 (2012학번, +3), 장기정 (수리과학과 2014학번, +3), 정성진 (2013학번, +3), 진우영 (수리과학과 2012학번, +3), 최두성 (수리과학과 2011학번, +3), 최인혁 (2015학번, +3), Muhammadfiruz Hassnov (2014학번, +3).

Let \(a\), \(b\) be distinct positive integers. Prove that there exists a prime \(p\) such that when dividing both \(a\) and \(b\) by \(p\), the remainder of \(a\) is less than the remainder of \(b\).

The best solution was submitted by 이종원 (2014학번). Congratulations!

Alternative solutions were submitted by 황성호 (+3), 정성진(+2), 박훈민 (+2). There were a few incorrect submissions (KSJ, JKJ, KDS, AHS, KKS, PKH).

Prove that there exist infinitely many pairs of positive integers \( (m, n) \) satisfying the following properties:

(1) gcd\( (m, n) = 1 \).

(2) \((x+m)^3 = nx\) has three distinct integer solutions.

The best solution was submitted by 이종원. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 김은혜 (+3), 김일희 (+3), 김찬민 (+3), 박훈민 (+3), 안현수 (+3), 어수강 (+3), 윤성철 (+3), 이영민 (+3), 장경석 (+3), 장기정 (+3), 정성진 (+3), 조준영 (+3), 채석주 (+3), 황성호 (+3), 박경호 (+2), 조남경 (+2). An incorrect solutions was submitted by N.J.H. (Some initials here might have been improperly chosen.)