# Tag Archives: 황성호 # 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: 2014-12 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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# 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: 2014-02 Series

Determine all positive integers $$\ell$$ such that $\sum_{n=1}^\infty \frac{n^3}{(n+1)(n+2)(n+3)\cdots (n+\ell)}$ converges and if it converges, then compute its value.

The best solution was submitted by 황성호 (2013학번). Congratulations!

Alternative solutions were submitted by 박훈민 (+3), 이종원 (+3), 채석주 (+3), 이영민 (+2), 조준영 (+2),정성진 (+3), 장기정 (+3), 오동우 (+3), 이상철 (+3), 어수강 (+3), 엄문용 (+3), 윤성철 (+3), 전한울 (+3), 박경호 (+2), 한대진 (+2), 서진솔 (+2), 이시우 (+2). Four incorrect solutions were submitted (J.K.S., N.J.H., A.H.S., C.J.H.).

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# Concluding Spring 2013

The top 5 participants of the semester are:

• 1st: 라준현 (08학번): 38 points
• 2nd: 서기원 (09학번): 29 points
• T-3rd: 김호진 (09학번): 25 points
• T-3rd: 황성호 (13학번): 25 points
• 5th: 김범수 (10학번): 19 points

Hearty congratulations to the prize winners! The prize ceremony will be held on Jun. 19 (Wed.) at 2PM.

We thank all of the participants for the nice solutions and your intereset you showed for POW. We hope to see you next semester with even better problems.

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# Solution: 2013-11 Integer coefficient complex-valued polynomials

Determine all polynomials $$P(z)$$ with integer coefficients such that, for any complex number $$z$$ with $$|z| = 1$$, $$| P(z) | \leq 2$$.

The best solution was submitted by 황성호, 13학번. Congratulations!

Another solution was submitted by 라준현(08학번, +3). Thank you for your participation.

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Let $F(x) = \sum_{n=1}^{1000} \cos (n^{3/2} x).$
Prove that $$F$$ has at least $$80$$ zeros in the interval $$(0, 2013)$$.