For \( \theta>0 \), let

\[

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

\]

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

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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!

Suppose that \( f : (2, \infty) \to (-2, 2) \) is a continuous function and there exists a positive constant \( m \) such that \( | 1 + xf(x) + (f(x))^2 | \leq m \) for any \( x > 2 \). Prove that, for any \( x > 2 \),

\[

\left| f(x) – \frac{\sqrt{x^2 -4}-x}{2} \right| \leq 6 \sqrt{m}.

\]

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

Here is his solution of problem 2017-05.

Alternative solutions were submitted by 위성군 (수리과학과 2015학번, +3), 조태혁 (수리과학과 2014학번, +3), 최인혁 (물리학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 오동우 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 김재현 (수리과학과 2016학번, +3), 김태균 (수리과학과 2016학번, +2).

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

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.