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.

For integer \( n \geq 1 \), define
\[
a_n = \sum_{k=0}^{\infty} \frac{k^n}{k!}, \quad b_n = \sum_{k=0}^{\infty} (-1)^k \frac{k^n}{k!}.
\]
Prove that \( a_n b_n \) is an integer.

The best solution was submitted by 황성호. Congratulations!

Similar solutions were submitted by 박훈민 (+3), 이규승 (+3), 이승훈 (+3), 이영민 (+3), 이종원 (+3), 장기정 (+3), 정성진 (+3), 채석주 (+3), Zhang Qiang (+2). Thank you for your participation.

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).

Let \(a_1,a_2,\ldots\) be an infinite sequence of positive real numbers such that \(\sum_{n=1}^\infty a_n\) converges. Prove that for every positive constant \(c\), there exists an infinite sequence \(i_1<i_2<i_3<\cdots\) of positive integers such that \(| i_n-cn^3| =O(n^2)\) and  \(\sum_{n=1}^\infty \left( a_{i_n} (a_1^{1/3}+a_2^{1/3}+\cdots+a_{i_n}^{1/3})\right)\) converges.

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

Alternative solutions were submitted by 정성진 (+3), 이종원 (+2), 이영민 (+2), 황성호 (+2), 김경석 (+2), 채석주 (+1). Incorrect solutions were submitted by B.H.J., P.K.H.

Suppose that \( a_1, a_2, \cdots \) are positive real numbers. Prove that
\[
\sum_{n=1}^{\infty} (a_1 a_2 \cdots a_n)^{1/n} \leq e \sum_{n=1}^{\infty} a_n \,.
\]

The best solution was submitted by 정성진. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 이영민 (+3), 이종원 (+3), 장기정 (+3), 정성진 (+3), 조준영 (+3), 황성호 (+2). Incorrect solutions were submitted by K.S.J., L.S.C. (Some initials here might have been improperly chosen.)