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.

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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.

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

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.

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}

\]

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).

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.

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 \(b\).

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.