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.

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).

Compute \[ \lim_{n\to \infty} \cos^{2016} (\pi\sqrt{n^2+4n+9}).\]

The best solution was submitted by Kang, Hanpil (강한필), 2016학번. Congratulations!

Here is his solution of problem 2016-8.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 이준호 (2016학번, +3), 장기정 (수리과학과 2014학번, +3), 유찬진 (수리과학과 2015학번, +3), 배형진 (마포고등학교 2학년, +3), 박기연 (2016학번, +3), 유현우 (한양대학교 화학공학과 2013학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 김재현 (2016학번, +3), 이예찬 (오송고등학교 교사, +3), 최백규 (2016학번, +3), 장창환 (기계공학과 2015학번, +3), 한대진 (인천예일중학교 교사, +3), 박은구 (연세대 수학과 대학원생, +3), 김태균 (2016학번, +3), 박정우 (한국과학영재학교 2016학번, +3), 이원웅 (건국대 수학과 2014학번, +3), 이상민 (수리과학과 2014학번, +3).

For a set \( A \subset \mathbb{R} \), let \( f(A) \) be the size of the largest set \( B \subset A \) such that \( (B+B) \cap B = \emptyset \). For a positive integer \( n \), let \( f(n) = \min_{0 \notin A, |A|=n} f(A) \). Prove that \( f(n) \geq n/3 \).

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

Here is his solution of problem 2016-7.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3).

Let \(A_1,A_2,\ldots,A_n\) be subsets of \(\{1,2,\ldots,n\}\) such that \(i\notin A_i\) for all \(i\). Prove that there exist four sets \(C_1,C_2,C_3,C_4\) such that \(C_1\cup C_2\cup C_3\cup C_4=\{1,2,\ldots,n\} \) and for all \(i\) and \(j\), if \(i\in C_j\), then \( \lvert A_i\cap C_j\rvert \le \frac12 \lvert A_i\rvert\).

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

Here is his solution of problem 2016-5.

Alternative solutions were submitted by 이준호 (2016학번, +2), 김경석 (연세대학교 의예과 2016학번, +2). An incorrect solution was received.

Note: There is a simpler solution.

Suppose that \( f \) is a real-valued convex function on \( \mathbb{R} \). Prove that the function \( X \mapsto \mathrm{Tr } f(X) \) on the vector space of \( N \times N \) Hermitian matrices is convex.

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

Here is his solution of problem 2016-6.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 이준호 (2016학번, +3), 장기정 (수리과학과 2014학번, +3), 유현우 (한양대학교 화학공학과 2013학번, +3), 이시우 (포항공대 수학과 2013학번, +3).

Let \(T\) be a tree on \(n\) vertices \(V=\{1,2,\ldots,n\}\). For two vertices \(i\) and \(j\), let \(d_{ij}\) be the distance between \(i\) and \(j\), that is the number of edges in the unique path from \(i\) to \(j\). Let \(D_T(x)=(x^{d_{ij}})_{i,j\in V}\) be the \(n\times n\) matrix. Prove that \[ \det (D_T(x))=(1-x^2)^{n-1}.\]

The best solution was submitted by Kim, Kee Tack (김기택, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-4.

Alternative solutions were submitted by 강한필 (2016학번, +3), 국윤범 (수리과학과 2015학번, +3), 김강식 (포항공대 수학과 2013학번, +3), 김경석 (연세대학교 의예과 2016학번, +3), 김동률 (수리과학과 2015학번, +3), 김재현 (2016학번, +3), 박기연 (2016학번, +3), 송교범 (서대전고등학교 3학년, +3), 이시우 (포항공대 수학과 2013학번, +3), 이종원 (수리과학과 2014학번, +3), 이준호 (2016학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), Muhammaadfiruz Hasanov (2014학번, +3), 김동규 (수리과학과 2015학번, +2), 김홍규 (수리과학과 2011학번, +2), 배형진 (마포고등학교 2학년, +2), 어수강 (서울대학교 수학교육과 박사과정, +2), 유찬진 (수리과학과 2015학번, +2), 윤준기 (전기및전자공학부 2014학번, +2), 이상민 (수리과학과 2014학번, +2), 이정환 (수리과학과 2015학번, +2).