# 2014-12 Rational ratios in a triangle

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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# Solution: 2014-11 Subsets of a countably infinite set

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).

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# 2014-11 Subsets of a countably infinite set

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.

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# Solution: 2014-10 Inequality with pi

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.

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# 2014-10 Inequality with pi

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

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# Solution: 2014-09 Product of series

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.

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# Solution: 2014-08 Two positive integers

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).

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# 2014-09 Product of series

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.

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# 2014-08 Two positive integers

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 $$b$$.

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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.