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.

For a positive integer \( n \), define \( f(n) \) by
\[
f(n) =
\begin{cases}
0 & \text{ if } n \equiv 0 \pmod{5} \\
1 & \text{ if } n \equiv \pm 1 \pmod{5} \\
-1 & \text{ if } n \equiv \pm 2 \pmod{5}
\end{cases}.
\]
Compute the infinite series
\[
\sum_{n=1}^{\infty} \frac{f(n)}{n} = 1 – \frac{1}{2} – \frac{1}{3} + \frac{1}{4} + \frac{1}{6} – \dots.
\]

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

Here is his solution of problem 2016-11.

Alternative solutions were submitted by 이상민 (수리과학과 2014학번, +3), 박정우 (한국과학영재학교 2016학번, +2), 윤준기 (전기및전자공학부 2014학번, +2), 최백규 (2016학번, +2).

Suppose that \( A \) is an \( n \times n \) matrix with integer entries and \( \det A = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k} \) for primes \( p_1, p_2, \dots, p_k \) and positive integers \( e_1, e_2, \dots, e_k \). Prove that there exist \( n \times n \) matrices \( B_1, B_2, \dots, B_k \) with integer entries such that \( A = B_1 B_2 \dots B_k \) and \( \det B_1 = p_1^{e_1}, \det B_2 = p_2^{e_2}, \dots, \det B_k = p_k^{e_k} \).

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

Here is his solution of problem 2016-10.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +2), 박정우 (한국과학영재학교 2016학번, +2).

Let \( A=(a_{ij})_{ij}\) be an \(n\times n\) matrix, where \[ a_{ij}=\begin{cases} 3 &\text{if }i=j,\\ (-1)^{\lvert i-j\rvert}&\text{otherwise.}\end{cases}\] Compute the determinant of \(A\).

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

Here is his solution of problem 2016-9.

Alternative solutions were submitted by 강한필 (2016학번, +3), 국윤범 (수리과학과 2015학번, +3), 김태균 (2016학번, +3), 박정우 (한국과학영재학교 2016학번, +3), 송교범 (서대전고등학교 3학년, +3), 이상민 (수리과학과 2014학번, +3), 이원웅 (건국대 수학과 2014학번, +3), 최백규 (2016학번, +3), 유찬진 (수리과학과 2015학번, +2), 윤준기 (전기및전자공학부 2014학번, +2).