For a positive integer \( n \), let \( d(n) \) be the number of positive divisors of \( n \). Prove that, for any positive integer \( M \), there exists a constant \( C>0 \) such that \( d(n) \geq C ( \log n )^M \) for infinitely many \( n \).

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

Here is his solution of problem 2016-21.

Alternative solutions were submitted by 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution), 국윤범 (수리과학과 2015학번, +3), 이상민 (수리과학과 2014학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이정환 (수리과학과 2015학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3), 조준영 (수리과학과 2012학번, +3), 이시우 (포항공대 수학과 2013학번, +3).

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.