# Concluding 2018 Spring

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

1st prize (Gold): Lee, Jongwon (이종원, 수리과학과 2014학번)
2nd prize (Silver): Chae, Jiseok (채지석, 수리과학 과 2016학번)
2nd prize (Silver): Han, Joon Ho (한준호,수리과학과 2015학번)
2nd prize (Silver): Lee, Bonwoo (이본우, 수리과학과 2017학번)
3rd prize (Bronze): Ko, Sunghun (고성훈, 2018학번)

이종원 (수리과학과 2014학번) 40/40
채지석 (수리과학과 2016학번) 35/40
한준호 (수리과학과 2015학번) 35/40
이본우 (수리과학과 2017학번) 32/40
고성훈 (2018학번) 20/40
김태균 (수리과학과 2016학번) 19/40
최인혁 (물리학과 2015학번) 10/40
김건우 (수리과학과 2017학번) 8/40
최백규 (생명과학과 2016학번) 6/40
하석민 (수리과학과 2017학번) 6/40
길현준 (2018학번) 3/40
강한필 (전산학부 2016학번) 3/40
문정욱 (2018학번) 3/40
노우진 (물리학과 2015학번) 1/40
윤정인 (물리학과 2016학번) 1/40

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# Solution: 2018-08 Large LCM

Let $$a_1$$, $$a_2$$, $$\ldots$$, $$a_m$$ be distinct positive integers. Prove that if $$m>2\sqrt{N}$$, then there exist $$i$$, $$j$$ such that the least common multiple of $$a_i$$ and $$a_j$$ is greater than $$N$$.

The best solution was submitted by Bonwoo Lee (이본우, 수리과학과 2017학번). Congratulations!

Here is his solution of problem 2018-08.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 김태균 (수리과학과 2016학번, +3), 한준호 (수리과학과 2015학번, +3), 이재우 (함양고등학교 3학년, +3).

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# Solution: 2018-01 Recurrence relation

Define a sequence $$\{ a_n \}$$ by $$a_1 = a$$ and
$a_n = \frac{2n-1}{n-1} a_{n-1} -1$
for $$n \geq 2$$. Find all real values of $$a$$ such that $$\lim_{n \to \infty} a_n$$ exists.

The best solution was submitted by Bonwoo Lee (이본우, 수리과학과 2017학번). Congratulations!

Here is his solution of problem 2018-01.

Alternative solutions were submitted by 강한필 (전산학부 2016학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이종원 (수리과학과 2014학번, +3), 채지석 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), 한준호 (수리과학과 2015학번, +3), 고성훈 (2018학번, +2), 김태균 (수리과학과 2016학번, +2), 송교범 (고려대 수학과 2017학번, +2), 이재우 (함양고등학교 3학년, +2), 노우진 (물리학과 2015학번) 및 윤정인 (물리학과 2016학번) (+2). Two incorrect solutions were received.

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# Solution: 2017-20 Convergence of a series

Determine whether or not the following infinite series converges. $\sum_{n=0}^{\infty} \frac{ 1 }{2^{2n}} \binom{2n}{n}.$

The best solution was submitted by Lee, Bonwoo (이본우, 2017학번). Congratulations!

Here is his solution of problem 2017-20.

Alternative solutions were submitted by 고성훈 (+3), 국윤범 (수리과학과 2015학번, +3), 길현준 (인천과학고등학교 2학년, +3), 김태균 (수리과학과 2016학번, +3), 민찬홍 (중앙대학교사범대학부속고등학교 3학년, +3), 유찬진 (수리과학과 2015학번, +3), 이원웅 (건국대 수학과 2014학번, +3), 이준협 (하나고등학교, +3), 채지석 (2016학번, +3), 최대범 (수리과학과 2016학번, +3), 하석민 (2017학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 이준성 (상문고등학교 1학년, +3), 정경훈 (서울대학교 컴퓨터공학과, +3), Mirali Ahmadili & Saba Dzmanashvili (2017학번, +3).

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# Concluding 2017 Spring

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

1st prize (Gold): Jo, Tae Hyouk (조태혁, 수리과학과 2014학번)
2nd prize (Silver): Huy Tùng Nguyễn (수리과학과 2016학번)
2nd prize (Silver): 최대범 (수리과학과 2016학번)
2nd prize (Silver): Lee, Bonwoo (이본우, 2017학번)
3rd prize (Bronze): Jang, Kijoung (장기정, 수리과학과 2014학번)

조태혁 (수리과학과 2014학번) 36/40
Huy Tung Nguyen (2016학번) 35/40
최대범 (수리과학과 2016학번) 31/40
이본우 (2017학번) 30/40
장기정 (수리과학과 2014학번) 26/40
위성군 (수리과학과 2015학번) 25/40
최인혁 (물리학과 2015학번) 25/40
오동우 (수리과학과 2015학번) 24/40
김태균 (수리과학과 2016학번) 20/40
Ivan Adrian Koswara (전산학부 2013학번) 12/40
강한필 (2016학번) 9/40
유찬진 (수리과학과 2015학번) 4/40
채지석 (2016학번) 3/40
곽상훈 (수리과학과 2013학번) 3/40
김재현 (수리과학과 2016학번) 3/40
이정환 (수리과학과 2015학번) 3/40
이준호 (2016학번) 3/40
홍혁표 (수리과학과 2013학번) 3/40
이태영 (수리과학과 2013학번) 2/40

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# Solution: 2017-10 An inequality for determinant

Let $$A$$, $$B$$ be matrices over the reals with $$n$$ rows. Let $$M=\begin{pmatrix}A &B\end{pmatrix}$$. Prove that $\det(M^TM)\le \det(A^TA)\det(B^TB).$

The best solution was submitted by Lee, Bonwoo (이본우, 17학번). Congratulations!

Here is his solution of problem 2017-10.

Alternative solutions were submitted by Huy Tung Nguyen (2016학번, +3), 조태혁 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +2). One incorrect solution was received.

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