Prove that for two non-zero complex numbers \(x\) and \(y\), if \(|x| ,| y|\le 1\), then \[ |x-y|\le |\log x-\log y|.\]

The best solution was submitted by Minjae Park (박민재), 수리과학과 2011학번. Congratulations!

Here is his solution of the problem 2014-19.

Alternative solutions were submitted by 박훈민 (수리과학과 2013학번, +3), 이병학 (2013학번, +3), 채석주 (2013학번, +2), 박지민 (2012학번, +3), 김범수 (2010학번, +3), 장기정 (2014학번, +3), 정경훈 (서울대 컴퓨터공학과 2006학번, +3, his solution), 윤성철 (홍익대학교 수학교육과, +3), 진형준 (인천대 2014학번, +2), 장유진 (홍익대학교 2013학번, +3), 정요한 (서울시립대학교 수학과, +3), 조현우 (경남과학고 3학년, +3).

Let \[p(x)=x^n+x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_1 x_1 + a_0\] be a polynomial. Prove that if \(p(z)=0\) for a complex number \(z\), then \[ |z| \le 1+ \sqrt{\max (|a_0|,|a_1|,|a_2|,\ldots,|a_{n-2}|)}.\]

The best solution was submitted by Minjae Park (박민재, 수리과학과 2011학번). Congratulations!

Here is his solution of the problem 2014-17.

An alternative solution was submitted by 조현우 (경남과학고등학교 3학년, +3).

For a (simple) graph \(G\), let \(o(G)\) be the number of odd-sized sets of pairwise non-adjacent vertices and let \(e(G)\) be the number of even-sized sets of pairwise non-adjacent vertices. Prove that if we can delete \(k\) vertices from \(G\) to destroy every cycle, then \[ | o(G)-e(G)|\le 2^{k}.\]

The best solution was submitted by Minjae Park (박민재, 수리과학과 2011학번). Congratulations!

Here is his solution.

An alternative solution was submitted by 김경석 (+3, 경기과학고 3학년). One incorrect solution was received (BHJ).

Let \( f(z) = z + e^{-z} \). Prove that, for any real number \( \lambda > 1 \), there exists a unique \( w \in H = \{ z \in \mathbb{C} : \text{Re } z > 0 \} \) such that \( f(w) = \lambda \).

The best solution was submitted by 박민재. Congratulations!

Similar solutions are submitted by 김동률(+3), 김범수(+3), 김호진(+3), 박지민(+3), 박훈민(+3), 양지훈(+3), 이시우(+3), 전한솔(+3), 정성진(+3), 조정휘(+3), 진우영(+3), Koswara(+3), Harmanto(+3). Thank you for your participation.

For real numbers \( a, b \), find the following limit.
\[
\lim_{n \to \infty} n \left( 1 – \frac{a}{n} – \frac{b \log (n+1)}{n} \right)^n.
\]

The best solution was submitted by 박민재. Congratulations!

Similar solutions are submitted by 김범수(+3), 박훈민(+3), 장경석(+3), 정성진(+3), 진우영(+3), 김홍규(+2), 박경호(+2). Thank you for your participation.

Let \(A=(a_{ij})\) be an \(n\times n\) upper triangular matrix such that \[a_{ij}=\binom{n-i+1}{j-i}\] for all \(i\le j\). Find the inverse matrix of \(A\).

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is his Solution of Problem 2012-20.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 이명재 (2012학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 박훈민 (대전과학고 2학년, +3), 윤성철 (홍익대학교 수학교육학과 2009학번, +3), 어수강 (서울대학교 수리과학부 석사과정, +3).

Let A be a finite set of complex numbers. Prove that there exists a subset B of A such that \[ \bigl\lvert\sum_{z\in B} z\bigr\lvert \ge \frac{ 1}{\pi}\sum_{z\in A} \lvert z\rvert.\]

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is Solution of Problem 2012-12.

Two incorrect solutions were submitted (M.J.L., W.S.J.).