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 \(\theta\) be a fixed constant. Characterize all functions \(f:\mathcal R\to \mathcal R\) such that \(f”(x)\) exists for all real \(x\) and for all real \(x,y\), \[ f(y)=f(x)+(y-x)f'(x)+ \frac{(y-x)^2}{2} f”(\theta y + (1-\theta) x).\]

The best solution was submitted by 장유진 (홍익대학교 수학교육과 2013학번). Congratulations!

Here is his solution of problem 2014-15.

Alternative solutions were submitted by 박민재 (수리과학과 2011학번, +3), 장기정 (2014학번, +2), 류상우 (서울대 수리과학부 2012학번, +2), 조현우 (경남과학고 3학년, +2), 윤성철 (홍익대학교 수학교육과, +2). (The most common mistake was to assume that if a Taylor series of an infinitely differentiable function f converges, then it converges to f.)

Let \(A\) and \(B\) be \(n\times n\) real matrices for an odd integer \(n\). Prove that if both \(A+A^T\) and \(B+B^T\) are invertible, then \(AB\neq 0\).

The best solution was submitted by Jimin Park (박지민, 전산학과 2012학번). Congratulations!

Here is his solution of problem 2014-18.

Alternative solutions were submitted by 채석주 (2013학번, +3), 정성진 (2013학번, +3), 장기정 (2014학번, +3), 박민재 (2011학번, +3), 김경석 (경기과학고등학교 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).

Prove or disprove that for all positive integers \(m\) and \(n\), \[ f(m,n)=\frac{2^{3(m+n)-\frac12} }{{\pi}} \int_0^{\pi/2} \sin^{ 2n – \frac12 }\theta \cdot \cos^{2m+\frac12}\theta \, d\theta\]  is an integer.

The best solution was submitted by 김경석 (경기과학고등학교 3학년). Congratulations!

Here is his solution.

Alternative solutions were submitted by 이병학 (2013학번, +2), 박훈민 (2013학번, +2), 배형진 (공항중학교 3학년, +2). One incorrect solution was submitted (LSC).

Prove that, for any unit vectors \( v_1, v_2, \cdots, v_n \) in \( \mathbb{R}^n \), there exists a unit vector \( w \) in \( \mathbb{R}^n \) such that \( \langle w, v_i \rangle \leq n^{-1/2} \) for all \( i = 1, 2, \cdots, n \). (Here, \( \langle \cdot, \cdot \rangle \) is a usual scalar product in \( \mathbb{R}^n \).)

The best solution was submitted by 어수강. Congratulations!

Alternative solutions were submitted by 이종원 (+3), 장기정 (+3), 정성진 (+3), 채석주 (+1), 황성호 (+1). Thank you for your participation.

Determine all triangles ABC such that all of \( \frac{AB}{BC}, \frac{BC}{CA}, \frac{CA}{AB}, \frac{\angle A}{\angle B}, \frac{\angle B}{\angle C}, \frac{\angle C}{\angle A}\) are rational.

The best solution was submitted by 황성호. Congratulations!

Alternative solutions were submitted by 정성진(+3), 이영민(+3), 채석주(+3), 이종원(+3), 장기정(+3), 배형진(+3), 남재현(+2), 김경민(+2), 박경호(+2), 서웅찬(+2). Thank you for your participation.

Prove or disprove that every uncountable collection of subsets of a countably infinite set must have two members whose intersection has at least 2014 elements.

The best solution was submitted by 장기정. Congratulations!

Alternative solutions were submitted by 이종원(+3), 정성진(+3), 채석주(+3), 황성호(+3), 김경석(+3), 어수강(+3). Two incorrect solutions were submitted (KKM, BHJ).

Prove that, for any sequences of real numbers \( \{ a_n \} \) and \( \{ b_n \} \), we have
\[
\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{a_m b_n}{m+n} \leq \pi \left( \sum_{m=1}^{\infty} a_m^2 \right)^{1/2} \left( \sum_{n=1}^{\infty} b_n^2 \right)^{1/2}
\]

The best solution was submitted by 장기정. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 김동석 (+3), 박경호 (+3), 이규승 (+3), 이영민 (+3), 이종원 (+3), 정성진 (+3), 채석주 (+3), 황성호 (+3), Zhang Qiang (+3). Thank you for your participation.