Solution: 2017-08 Long arithmetic progression

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).

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Solution: 2017-07 Supremum of a series

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).

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2017-08 Long arithmetic progression

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 \(\lvert S\rvert >\varepsilon n\)?

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Solution: 2017-06 Powers of 2

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).

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Midterm break

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!

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Solution: 2017-05 Inequality for a continuous function

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).

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