# Monthly Archives: June 2014 # 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: 2014-13 Unit vectors

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.

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# Solution: 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.

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.

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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 $$\mathbb{R}^n$$.)