# Concluding 2014 Spring

Thanks all for participating POW actively. Here’s the list of winners:

• 1st prize (Gold): Lee, Jongwon (이종원) – 2014학번
• 2nd prize (Silver): Jeong, Seongjin (정성진) – 수리과학과 2013학번
• 2nd prize (Silver): Jang, Kijoung (장기정) – 2014학번
• 4th prize: Hwang, Sungho (황성호) – 수리과학과 2013학번
• 5th prize: Chae, Seok Joo (채석주) – 수리과학과 2013학번

이종원 40
정성진 39
장기정 39
황성호 38
채석주 29
이영민 25
박훈민 18
조준영 17
김경석 17
어수강 16
박경호 15
윤성철 9
장경석 9
김일희 8
안현수 6
오동우 6
정진야 6
이규승 6
Zhang Qiang 5
이시우 5
한대진 5
남재현 5
김범수 4
김정민 4
권현우 3
김동석 3
김은혜 3
김찬민 3
엄문용 3
이상철 3
이주호 3
전한울 3
심병수 3
이승훈 3
배형진 3
서진솔 2
조남경 2
김경민 2
서웅찬 2

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# Solution: 2014-06 Inequality with e

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.)

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# Solution: 2013-19 Integral inequality

Suppose that a function $$f:[0, 1] \to (0, \infty)$$ satisfies that
$\int_0^1 f(x) dx = 1.$
Prove the following inequality.
$\left( \int_0^1 |f(x)-1| dx \right)^2 \leq 2 \int_0^1 f(x) \log f(x) dx.$

The best solution was submitted by 정성진. Congratulations!

Similar solutions are submitted by 박민재(+3), 진우영(+3). Thank you for your participation.

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Let $$x, y$$ be real numbers satisfying $$y \geq x^2 + 1$$. Prove that there exists a bounded random variable $$Z$$ such that
$E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y.$
Here, $$E$$ denotes the expectation.