For \( \theta>0 \), let
\[
f(\theta) = \sum_{n=1}^{\infty} \left( \frac{1}{n+ \theta} – \frac{1}{n+ 3\theta} \right).
\]
Find \( \sup_{\theta > 0} f(\theta) \).

The best solution was submitted by Oh, Dong Woo (오동우, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2017-07.

Alternative solutions were submitted by 조태혁 (수리과학과 2014학번, +3), 위성군 (수리과학과 2015학번, +3), 최인혁 (물리학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), Huy Tung Nguyen (2016학번, +3), 이본우 (2017학번, +3), 김태균 (수리과학과 2016학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3).

Does there exist infinitely many positive integers \(n\) such that the first digit of \(2^n\) is \(9\)?

The best solution was submitted by  Jo, Tae Hyouk (조태혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2017-06.

Alternative solutions were submitted by 강한필 (2016학번, +3, solution), 김태균 (수리과학과 2016학번, +3), 오동우 (수리과학과 2015학번, +3), 위성군 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 장기정 (수리과학과 2014학번, +3, solution), 채지석 (2016학번, +3), 최대범 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), Huy Tung Nguyen (2016학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3), Saba Dzmanashvili (+3).

Prove (or disprove) that exactly one of the following is true for every subset \(A\) of \(\{ (i,j): i,j\in\{1,2,\ldots,n\}, i\neq j\}\).

(i) There exists a sequence of distinct integers \(i_1,i_2,\ldots,i_k\in \{1,2,\ldots,n\}\) for some integer \(k>1\) such that \( (i_1,i_2), (i_2,i_3),\ldots,(i_{k-1},i_k), (i_k,i_1)\in A\).

(ii) There exists a collection of finite sets \( A_1,A_2,\ldots,A_n\) such that for all distinct \(i,j\in\{1,2,\ldots,n\}\), \((i,j)\in A\) if and only if \( \lvert A_i\cap A_j\rvert > \frac12 \lvert A_i\rvert \) and \( \lvert A_i\cap A_j\rvert \le  \frac12 \lvert A_j\rvert \)

The best solution was submitted by Jo, Tae Hyouk (조태혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2017-4.

Alternative solutions were submitted by 강한필 (2016학번, +3), 김태균 (수리과학과 2016학번, +3), 배형진 (마포고 3학년, +3), 오동우 (수리과학과 2015학번, +3), 위성군 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번), Ivan Adrian Koswara (전산학부 2013학번, +3), 송교범 (고려대 수학과 2017학번, +2), 조정휘 (건국대학교 수학과 2014학번, +2), Huy Tung Nguyen (2016학번, +2).

Reference: Lai, Endrullis, and Moss, Majority Digraphs, Proc. Amer. Math. Soc. 144 (2016), 3701-3715.

For an integer \( n \geq 4 \), find the solutions of the equation
\[
\sum_{k=1}^n \frac{\sin \frac{k\pi}{n+1}}{\sin (\frac{k\pi}{n+1} -x)} = 0.
\]

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

Here is his solution of problem 2017-03.

Alternative solutions were submitted by 위성군 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), Huy Tung Nguyen (2016학번, +3), 조정휘 (건국대학교 수학과 2014학번, +3), 배형진 (마포고 3학년, +2), 오동우 (수리과학과 2015학번, +2).

Let \(a_1,a_2,\ldots,a_n\) be distinct points in \(\mathbb R^4\). Does there exist a non-zero polynomial \(P(x_1,x_2,x_3,x_4)\) such that
(1) the degree of \(P\) is at most \(\lceil\sqrt{5} n^{1/4}\rceil\) and
(2) \(P(a_i)=0\) for all \(i=1,2,\ldots,n\)?

The best solution was submitted by You, Chanjin (유찬진, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2017-02.

Alternative solutions were submitted by 김태균 (수리과학과 2016학번, +3), 박지민 (전산학부 박사 2017학번, +3), 배형진 (마포고 3학년, +3), 송교범 (고려대 수학과 2017학번, +3), 오동우 (수리과학과 2015학번, +3), 위성군 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이준호 (2016학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), 홍혁표 (수리과학과 2013학번, +3), Huy Tung Nguyen (2016학번, +3).

Let \( A, B, C \) be \( N \times N \) Hermitian matrices with \( C = A+B \). Let \( \alpha_1 \geq \dots \geq \alpha_N \), \( \beta_1 \geq \dots \geq \beta_N \), \( \gamma_1 \geq \dots \geq \gamma_N \) be the eigenvalues of \( A, B, C \), respectively. For any \( 1 \leq k \leq N \), prove that
\[ \gamma_1 + \gamma_2 + \dots + \gamma_k \leq (\alpha_1 + \alpha_2 + \dots + \alpha_k) + (\beta_1 + \beta_2 + \dots + \beta_k) \]

The best solution was submitted by Sounggun Wee (위성군, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2017-01.

Alternative solutions were submitted by 강한필 (2016학번, +3), 김태균 (수리과학과 2016학번, +3), 배형진 (마포고 3학년, +3), 오동우 (수리과학과 2015학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이정환 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), Huy Tung Nguyen (2016학번, +3), 곽상훈 (수리과학과 2013학번, +3), 이본우 (2017학번, +3), 이태영 (수리과학과 2013학번, +2).

Suppose that \( z_1, z_2, \dots, z_n \) are complex numbers satisfying \( \sum_{k=1}^n z_k = 0 \). Prove that
\[
\sum_{k=1}^n |z_{k+1} – z_k|^2 \geq 4 \sin^2 \left( \frac{\pi}{n} \right) \sum_{k=1}^n |z_k|^2,
\]
where we let \( z_{n+1} = z_1 \).

The best solution was submitted by Kim, Taegyun (김태균, 2016학번). Congratulations!

Here is his solution of problem 2016-23.

Alternative solutions were submitted by 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution), 국윤범 (수리과학과 2015학번, +3), 김기현 (수리과학과 대학원생, +3, alternative solution), 이상민 (수리과학과 2014학번, +2). One incorrect solution was submitted.

Let \(M_n=(a_{ij})_{ij}\) be an \(n\times n\) matrix such that \[a_{ij}=\binom{2(i+j-1)}{i+j-1}.\] What is \(\det M_n\)?

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

Here is his solution of problem 2016-22.

Alternative solutions were submitted by 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 이상민 (수리과학과 2014학번, +3), 채지석 (2016학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 홍진표 (서울대학교 재료공학부 2013학번, +3).