Find the value of

\[

\sum_{n=1}^{\infty} \frac{1+ \frac{1}{2} + \dots + \frac{1}{n}}{n(2n-1)}.

\]

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Find the value of

\[

\sum_{n=1}^{\infty} \frac{1+ \frac{1}{2} + \dots + \frac{1}{n}}{n(2n-1)}.

\]

Find all positive integers \( a, b, c \) satisfying \[3^a + 5^b = 2^c.\]

The best solution was submitted by Huy Tùng Nguyễn (2016학번). Congratulations!

Here is his solution of problem 2017-09.

Alternative solutions were submitted by 조태혁 (수리과학과 2014학번, +3), 이본우 (2017학번, +3), 최대범 (수리과학과 2016학번, +2), 이재우 (함양고등학교 2학년, +2).

Let \(A\), \(B\) be matrices over the reals with \(n\) rows. Let \(M=\begin{pmatrix}A &B\end{pmatrix}\). Prove that \[ \det(M^TM)\le \det(A^TA)\det(B^TB).\]

Find all positive integers \( a, b, c \) satisfying

\[

3^a + 5^b = 2^c.

\]

Does there exist a constant \(\varepsilon>0\) such that for each positive integer \(n\) and each subset \(A\) of \(\{1,2,\ldots,n\}\) with \(\lvert A\rvert<\varepsilon n\), there exists an artihmetic progression \(S\) in \(\{1,2,\ldots,n\}\) such that \( S\cap A=\emptyset\) and \(\lvert S\rvert >\varepsilon n\)?

The best solution was submitted by Huy Tùng Nguyễn (2016학번). Congratulations!

Here is his solution of problem 2017-8.

Alternative solutions were submitted by 조태혁 (수리과학과 2014학번, +3), 위성군 (수리과학과 2015학번, +3), 최인혁 (물리학과 2015학번, +3, solution), 오동우 (수리과학과 2015학번, +3), 최대범 (수리과학과 2016학번, +3), 이본우 (2017학번, +3), 김태균 (수리과학과 2016학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3), 이재우 (함양고등학교 2학년, +3), 장기정 (수리과학과 2014학번, +2).

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 a constant \(\varepsilon>0\) such that for each positive integer \(n\) and each subset \(A\) of \(\{1,2,\ldots,n\}\) with \(\lvert A\rvert<\varepsilon n\), there exists an artihmetic progression \(S\) in \(\{1,2,\ldots,n\}\) such that \( S\cap A=\emptyset\) and \(\lvert S\rvert >\varepsilon n\)?

For \( \theta>0 \), let

\[

f(\theta) = \sum_{n=1}^{\infty} \left( \frac{1}{n+ \theta} – \frac{1}{n+ 3\theta} \right).

\]

Find \( \sup_{\theta > 0} f(\theta) \).

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).

The problem of the week will take a break during the midterm exam period and return on April 28, Friday. Good luck on your midterm exams!