# 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: 2018-12 Property of Eigenvectors

Let $$A$$ be a $$2\times 2$$ matrix. Prove that if $$Av_1=\lambda_1v_1$$ and $$Av_2=\lambda_2v_2$$ for distinct reals $$\lambda_1$$ and $$\lambda_2$$ and nonzero vectors $$v_1$$ and $$v_2$$, then both columns of $$A-\lambda_1 I$$ is a multiple of $$v_2$$.

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2018-12.

Alternative solutions were submitted by 고성훈 (2018학번, +3), 권홍 (중앙대 물리학과, +3), 길현준 (2018학번, +3), 김태균 (수리과학과 2016학번, +3), 이본우 (수리과학과 2017학번, +3), 채지석 (수리과학과 2016학번, +3), 한준호 (수리과학과 2015학번, +3).

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# Solution: 2018-09 Sum of digits

For a positive integer $$n$$, let $$S(n)$$ be the sum of all decimal digits in $$n$$, i.e., if $$n = n_1 n_2 \dots n_m$$ is the decimal expansion of $$n$$, then $$S(n) = n_1 + n_2 + \dots + n_m$$. Find all positive integers $$n$$ and $$r$$ such that $$(S(n))^r = S(n^r)$$.

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2018-09.

Alternative solutions were submitted by 채지석 (수리과학과 2016학번, +3), 한준호 (수리과학과 2015학번, +3), 이본우 (수리과학과 2017학번, +3), 권홍 (중앙대 물리학과, +2).

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# Solution: 2018-07 A tridiagonal matrix

Let $$S$$ be an $$(n+1) \times (n+1)$$ matrix defined by
$S_{ij} = \begin{cases} (n+1)-i & \text{ if } j=i+1, \\ i-1 & \text{ if } j=i-1, \\ 0 & \text{ otherwise. } \end{cases}$
Find all eigenvalues of $$S$$.

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2018-07.

Alternative solutions were submitted by 한준호 (수리과학과 2015학번, +3), 채지석 (수리과학과 2016학번, +3), Hitesh Kumar (Imperial College London, +2), 고성훈 (2018학번, +2).

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# Solution: 2018-04 An inequality

Let $$x_1,x_2,\ldots,x_n$$ be reals such that $$x_1+x_2+\cdots+x_n=n$$ and $$x_1^2+x_2^2+\cdots +x_n^2=n+1$$. What is the maximum of $$x_1x_2+x_2x_3+x_3x_4+\cdots + x_{n-1}x_n+x_nx_1$$?

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2018-04.

Alternative solutions were submitted by 이본우 (수리과학과 2017학번, +3), 채지석 (수리과학과 2016학번, +3), 고성훈 (2018학번, +2).

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# Concluding 2016 Fall

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

1st prize (Gold): Shin, Joonhyung (신준형, 수리과학과 2015학번)
2nd prize (Silver): Jang, Kijoung (장기정, 수리과학과 2014학번).
2nd prize (Silver): Kim, Taegyun (김태균, 수리과학과 2014학번).
2nd prize (Silver): Kook, Yun Bum (국윤범, 수리과학과 2015학번).
3rd prize (Bronze): Lee, Sangmin (이상민, 수리과학과 2014학번).
3rd prize (Bronze): Lee, Jongwon (이종원, 수리과학과 2014학번).

신준형 (수리과학과 2015학번) 32, 장기정 (수리과학과 2014학번) 31, 김태균 (2016학번) 30, 국윤범 (수리과학과 2015학번) 29, 이상민 (수리과학과 2014학번) 19, 이종원 (수리과학과 2014학번) 19, 최대범 (2016학번) 16, 윤준기 (전기및전자공학부 2014학번) 14, 최인혁 (물리학과 2015학번) 13, 채지석 (2016학번) 12, 김재현 (2016학번) 11, 이정환 (수리과학과 2015학번) 9, Ivan Adrian Koswara (전산학부 2013학번) 6, 강한필 (2016학번) 6, 위성군 (수리과학과 2015학번) 6, 김기택 (수리과학과 2015학번) 6, 박기연 (2016학번) 5, 한준호 (수리과학과 2015학번) 5, 조준영 (수리과학과 2012학번) 3, 박현준 (물리학과 2014학번) 3, 오동우 (2015학번) 3, 유찬진 (수리과학과 2015학번) 3, 임성혁 (2016학번) 3, Muhammaadfiruz Hasanov (2014학번) 3, 정의현 (수리과학과 2015학번) 2, 박진호 (물리학과 2015학번) 2, 정성진 (수리과학과 2013학번) 2.

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# Solution: 2016-13 How to divide camels

A rich old man had $$k$$ sons and $$N$$ camels in the herd. The will of the father stated that his $$r$$-th son should receive $$1/ N_r$$ of his camels for $$r = 1, 2, \dots, k$$. Since $$N+1$$ is a common multiple of $$N_1, N_2, \dots, N_k$$, the sons could not divided $$N$$ camels as their father wished. The sons visited a wise man to solve the issue. The wise man listened about the will, and he brought his own camel, which he added to the herd. The herd was then divided up according to the old man’s wishes. The wise man then took back the one camel that remained, which was his own. For given $$k$$, find the maximal number of camels $$N \equiv N(k)$$ for which there is a solution to the problem where $$N_1, N_2, \dots, N_k$$ are positive integers.

The best solution was submitted by Jongwon Lee (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2016-13.

Alternative solutions were submitted by 최인혁 (물리학과 2015학번, +3), 국윤범 (수리과학과 2015학번, +3), 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (2016학번, +3), 김재현 (2016학번, +2), 김태균 (2016학번, +2), 한준호 (수리과학과 2015학번, +2), 이상민 (수리과학과 2014학번, +2). One incorrect solution was submitted.

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# Concluding 2016 Spring

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

1st prize (Gold): Kook, Yun Bum (국윤범, 수리과학과 2015학번)
2nd prize (Silver): Jang, Kijoung (장기정, 수리과학과 2014학번).
3rd prize (Bronze): Lee, Sangmin (이상민, 수리과학과 2014학번)
3rd prize (Bronze): Lee, Jongwon (이종원, 수리과학과 2014학번).
3rd prize (Bronze): Lee, Junho (이준호, 2016학번).

국윤범 (수리과학과 2015학번), 장기정 (수리과학과 2014학번), 이상민 (수리과학과 2014학번), 이종원 (수리과학과 2014학번), 이준호 (2016학번), 강한필 (2016학번), 유찬진 (수리과학과 2015학번), 윤준기 (전기및전자공학부 2014학번), Muhammaadfiruz Hasanov (2014학번), 김동규 (수리과학과 2015학번), 최백규 (2016학번), 김기택 (수리과학과 2015학번), 조태혁 (수리과학과 2014학번), 김동률 (수리과학과 2015학번), 김태균 (2016학번), 박기연 (2016학번), 최대범 (2016학번), 이정환 (수리과학과 2015학번), 김강식 (포항공대 수학과 2013학번), 김동하 (기계공학과 2014학번), 김재현 (2016학번), 이태영 (2013학번), 장창환 (기계공학과 2015학번), 정성진 (수리과학과 2013학번), 최인혁 (물리학과 2015학번), 김홍규 (수리과학과 2011학번), 노희광 (화학과 2014학번), 안현수 (2016학번), 홍혁표 (수리과학과 2013학번).

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# Solution: 2016-1 Flipping Signs

Prove that for every $$x_1, x_2,\ldots,x_n\in [0,1]$$, there exist $$\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_n\in\{1/2,-1/2\}$$ such that for all $$k=1,2,\ldots,n-1$$, $\left\lvert \sum_{i=1}^k \varepsilon_i x_i-\sum_{i=k+1}^n \varepsilon_i x_i \right\rvert\le 1.$

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2016-1.

Alternative solutions were submitted by 노희광 (화학과 2014학번, +2), 안현수 (2016학번, +2), 이상민 (수리과학과 2014학번, +2), 홍혁표 (수리과학과 2013학번, +2). There were 10 incorrect submissions.

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# Concluding 2015 Fall

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

1st prize (Gold): Lee, Jongwon (이종원), 수리과학과 2014학번.
2nd prize (Silver): Park, Sunghyuk (박성혁), 수리과학과 2014학번.
3rd prize (Bronze): Shin, Joonhyung (신준형), 2015학번.
3rd prize (Bronze): Jang, Kijoung (장기정), 수리과학과 2014학번.
3rd prize (Bronze): Choi, Inhyeok (최인혁), 2015학번.

이종원 (수리과학과 2014학번) 37점, 박성혁 (수리과학과 2014학번) 36점, 신준형 (2015학번) 33점, 장기정 (수리과학과 2014학번) 32점, 최인혁 (2015학번) 32점, 이영민 (수리과학과 2012학번) 18점, 박훈민 (수리과학과 2013학번) 17점, 김동률 (2015학번) 10점, 이상민 (수리과학과 2014학번) 8점, 김재준 (2014학번) 6점, 이정환 (2015학번) 6점, 오동우 (2015학번) 5점, 유찬진 (2015학번) 5점, 함도규 (2015학번) 5점, 이신영 (물리학과 2012학번) 4점, 김경석 (2015학번) 3점, 김기택 (2015학번) 3점, 김희주 (2015학번) 2점, 이호일 (수리과학과 2013학번) 2점,  이경훈 (수리과학과 2014학번) 1점.

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