Find the value of
\[
\sum_{n=1}^{\infty} \frac{1+ \frac{1}{2} + \dots + \frac{1}{n}}{n(2n-1)}.
\]

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

Here is his solution of problem 2017-11.

Alternative solutions were submitted by Huy Tung Nguyen (2016학번, +3), 최대범 (수리과학과 2016학번, +3), 이본우 (2017학번, +2).

This was the last problem of Spring 2017. Thank you for participating POW actively.

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.

Let \( G \) be a subgroup of \( GL_2 (\mathbb{R}) \) generated by \( \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} \) and \( \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \). Let \( H \) be a subset of \( G \) that consists of all matrices in \( G \) whose diagonal entries are \( 1 \). Prove that \( H \) is a subgroup of \( G \) but not finitely generated.

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

Here is his solution of problem 2016-3.

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