Let \(V_1,V_2,\ldots\) be countably many \(k\)-dimensional subspaces of \(\mathbb{R}^n\). Prove that there exists an \((n-k)\)-dimensional subspace \(W\) of \(\mathbb{R}^n\) such that \(\dim V_i\cap W=0\) for all \(i\).

The best solution was submitted by Shin, Joonhyung (신준형, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-20.

Alternative solutions were submitted by 김태균 (2016학번, +3), 국윤범 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution). One incorrect solution was submitted.

Let
\[
P(k) = \sum_{i_1=1}^{\infty} \dots \sum_{i_k=1}^{\infty} \frac{1}{i_1 \dots i_k (i_1 + \dots + i_k)}
\]
for a positive integer \( k \). Find \( \zeta(k+1) / P(k) \), where \( \zeta \) is the Riemann-zeta function.

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

Here is his solution of problem 2016-19.

Alternative solutions were submitted by 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3), 김태균 (2016학번, +3).

Suppose that we have a list of \(2n+1\) integers such that whenever we remove any one of them, the remaining can be partitioned into two lists of \(n\) integers with the same sum. Prove that all \(2n+1\) integers are equal.

The best solution was submitted by Joonhyung Shin (신준형, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-18.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3, solution), 김태균 (2016학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 김재현 (2016학번, +3), 채지석 (2016학번, +3), 강한필 (2016학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3), 김기현 (수리과학과 대학원생, +3). One incorrect solution was received.

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 \([-2,2]\).

The best solution was submitted by Choi, Daebeom (최대범, 2016학번). Congratulations!

Here is his solution of problem 2016-17. (There are a few typos.)

No alternative solutions were submitted.

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.