Solution: 2016-16 Column spaces

Let \(A\) be a square matrix with real entries such that \[ A A^T+A^T A = A+A^T.\] Prove that \(A\) and \(A^T\) have the same column space.

The best solution was submitted by Koon, Yun Bum (국윤범, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-16.

Alternative solutions were submitted by 장기정 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3), 최인혁 (물리학과 2015학번, +3), 신준형 (수리과학과 2015학번, +3), 최대범 (2016학번, +3), 김태균 (2016학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 위성군 (수리과학과 2015학번, +3), 이정환 (수리과학과 2015학번, +3). Two incorrect solutions were received.

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2016-17 Integral with two variables

Set \[ L(z,w)=\int_{-2}^2\int_{-2}^2 ( \log(z-x)-\log(z-y))( \log(w-x)-\log(w-y))Q(x,y) dx dy, \]
for \(z,w\in \mathbb{C}\setminus(-\infty, 2] \), where \[ Q(x,y)= \frac{4-xy}{(x-y)^2\sqrt{4-x^2}\sqrt{4-y^2}}. \]
Prove that \[ L(z,w)=2\pi^2 \log \left[ \frac{(z+R(z))(w+R(w))}{2(zw-4+R(z)R(w))} \right], \]
where \(R(z)=\sqrt{z^2-4}\) with branch cut \([-2,2]\).

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Solution: 2016-15 Pair of integers

Find all pairs of positive integers \( a \) and \( b \) such that \( a | (b^2 + b + 1) \) and \( b | (a^2 + a + 1) \).

The best solution was submitted by Kijoung Jang (장기정, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2016-15.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김강식 (포항공대 수학과 2013학번, +3), 김기택 (수리과학과 2015학번, +3), 김재현 (2016학번, +3), 김태균 (2016학번, +3), 박기연 (2016학번, +3), 박찬우 (서울대학교 통계학과 2016학번, +3), 신준형 (수리과학과 2015학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이상민 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3, alternative solution), 채지석 (2016학번, +3), 최대범 (2016학번, +3), 최인혁 (물리학과 2015학번, +3), 박현준 (물리학과 2014학번, +3), 위성군 (수리과학과 2015학번, +3), 김영헌 (개성고등학교 3학년, +3), Saba Dzmanashvili (+3), Jonathan French (+2), 정의현 (수리과학과 2015학번, +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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Solution: 2016-12 A series

Let \(k\) be a positive integer. Let \(a_n=1\) if \(n\) is not a multiple of \(k+1\), and \(a_n=-k\) if \(n\) is a multiple of \(k+1\). Compute \[\sum_{n=1}^\infty \frac{a_n}{n}.\]

The best solution was submitted by Choi, Inhyeok (최인혁, 물리학과 2015학번). Congratulations!

Here is his solution of problem 2016-12.

Alternative solutions were submitted by 강한필 (2016학번, +3), 국윤범 (수리과학과 2015학번, +3), 김기택 (수리과학과 2015학번, +3), 김재현 (2016학번, +3), 김태균 (2016학번, +3), 김태형 (EEWS대학원 석사과정, +3), 박찬우 (서울대학교 통계학과 2016학번, +3), 신준형 (수리과학과 2015학번, +3), 오동우 (2015학번, +3), 유찬진 (수리과학과 2015학번, +3), 이정환 (수리과학과 2015학번, +3), 이종원 (수리과학과 2014학번, +3), 임성혁 (2016학번, +3), 장기정 (수리과학과 2014학번, +3), 채지석 (2016학번, +3), 최대범 (2016학번, +3), 한준호 (수리과학과 2015학번, +3), Muhammaadfiruz Hasanov (2014학번, +3), 박기연 (2016학번, +2), 박진호 (물리학과 2015학번, +2), 송교범 (서대전고등학교 3학년, +2), 송민학 (월촌중학교 3학년, +2), 윤준기 (전기및전자공학부 2014학번, +2), 정성진 (수리과학과 2013학번, +2), 이본우 (대구과학고등학교 3학년, +2), 이상민 (수리과학과 2014학번, +2), 이시우 (포항공대 수학과 2013학번, +2). There were 2 incorrect solutions and 2 submissions by email missing attachments.

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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.

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2016-12 A series

Let \(k\) be a positive integer. Let \(a_n=1\) if \(n\) is not a multiple of \(k+1\), and \(a_n=-k\) if \(n\) is a multiple of \(k+1\). Compute \[\sum_{n=1}^\infty \frac{a_n}{n}.\]

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