Concluding 2017 Spring

Thanks all for participating POW actively. Here’s the list of winners:

1st prize (Gold): Jo, Tae Hyouk (조태혁, 수리과학과 2014학번)
2nd prize (Silver): Huy Tùng Nguyễn (수리과학과 2016학번)
2nd prize (Silver): 최대범 (수리과학과 2016학번)
2nd prize (Silver): Lee, Bonwoo (이본우, 2017학번)
3rd prize (Bronze): Jang, Kijoung (장기정, 수리과학과 2014학번)

조태혁 (수리과학과 2014학번) 36/40
Huy Tung Nguyen (2016학번) 35/40
최대범 (수리과학과 2016학번) 31/40
이본우 (2017학번) 30/40
장기정 (수리과학과 2014학번) 26/40
위성군 (수리과학과 2015학번) 25/40
최인혁 (물리학과 2015학번) 25/40
오동우 (수리과학과 2015학번) 24/40
김태균 (수리과학과 2016학번) 20/40
Ivan Adrian Koswara (전산학부 2013학번) 12/40
강한필 (2016학번) 9/40
유찬진 (수리과학과 2015학번) 4/40
채지석 (2016학번) 3/40
곽상훈 (수리과학과 2013학번) 3/40
김재현 (수리과학과 2016학번) 3/40
이정환 (수리과학과 2015학번) 3/40
이준호 (2016학번) 3/40
홍혁표 (수리과학과 2013학번) 3/40
이태영 (수리과학과 2013학번) 2/40

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Solution: 2017-11 Infinite series

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.

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Solution: 2017-10 An inequality for determinant

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

The best solution was submitted by Lee, Bonwoo (이본우, 17학번). Congratulations!

Here is his solution of problem 2017-10.

Alternative solutions were submitted by Huy Tung Nguyen (2016학번, +3), 조태혁 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +2). One incorrect solution was received.

 

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Solution: 2017-09 A Diophantine Equation

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).

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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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