2016-1 Flipping signs

Prove that for every \( x_1, x_2,\ldots,x_n\in [0,1]\), there exist \(\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_n\in\{1/2,-1/2\}\) such that for all \(k=1,2,\ldots,n-1\), \[ \left\lvert \sum_{i=1}^k \varepsilon_i x_i-\sum_{i=k+1}^n \varepsilon_i x_i \right\rvert\le 1.\]

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Concluding 2015 Fall

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

1st prize (Gold): Lee, Jongwon (이종원), 수리과학과 2014학번.
2nd prize (Silver): Park, Sunghyuk (박성혁), 수리과학과 2014학번.
3rd prize (Bronze): Shin, Joonhyung (신준형), 2015학번.
3rd prize (Bronze): Jang, Kijoung (장기정), 수리과학과 2014학번.
3rd prize (Bronze): Choi, Inhyeok (최인혁), 2015학번.

이종원 (수리과학과 2014학번) 37점, 박성혁 (수리과학과 2014학번) 36점, 신준형 (2015학번) 33점, 장기정 (수리과학과 2014학번) 32점, 최인혁 (2015학번) 32점, 이영민 (수리과학과 2012학번) 18점, 박훈민 (수리과학과 2013학번) 17점, 김동률 (2015학번) 10점, 이상민 (수리과학과 2014학번) 8점, 김재준 (2014학번) 6점, 이정환 (2015학번) 6점, 오동우 (2015학번) 5점, 유찬진 (2015학번) 5점, 함도규 (2015학번) 5점, 이신영 (물리학과 2012학번) 4점, 김경석 (2015학번) 3점, 김기택 (2015학번) 3점, 김희주 (2015학번) 2점, 이호일 (수리과학과 2013학번) 2점,  이경훈 (수리과학과 2014학번) 1점.

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Solution: 2015-24 Hölder inequality for matrices

Let \( A, B \) are \( n \times n \) Hermitian matrices and \( p, q \in [1, \infty] \) with \( \frac{1}{p} + \frac{1}{q} = 1 \). Prove that
\[
| Tr (AB) | \leq \| A \|_{S^p} \| B \|_{S^q}.
\]
(Here, \(\| A \|_{S^p} \) is the \(p\)-Schatten norm of \( A \), defined by
\[
\| A \|_{S^p} = \left( \sum_{i=1}^n |\lambda_i|^p \right)^{1/p},
\]
where \( \lambda_1, \lambda_2, \dots, \lambda_n \) are the eigenvalues of \( A \).)

The best solution was submitted by Park, Sunghyuk (박성혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-24.

Alternative solutions were submitted by 신준형 (2015학번, +3), 이정환 (2015학번, +3), 이종원 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3), 김동률 (2015학번, +2), 최인혁 (2015학번, +2).

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Solution: 2015-23 Fixed points

Let \(f:[0,1)\to[0,1)\)  be a function such that \[ f(x)=\begin{cases} 2x,&\text{if }0\le 2x\lt 1,\\ 2x-1, & \text{if } 1\le 2x\lt 2.\end{cases}\] Find all \(x\) such that \[ f(f(f(f(f(f(f(x)))))))=x.\]

The best solution was submitted by Park, Sunghyuk (박성혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-23.

Alternative solutions were submitted by 김동률 (2015학번, +3), 신준형 (2015학번, +3), 유찬진 (2015학번, +3), 이상민 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3, his solution), 장기정 (수리과학과 2014학번, +3), 정호진 (동북고등학교 2학년, +3), 최인혁 (2015학번, +3), Daulet Kurmantayev (?, +3), 최동준 (포항공대 수학과 2013학번, +2).

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2015-24 Hölder inequality for matrices

Let \( A, B \) are \( n \times n \) Hermitian matrices and \( p, q \in [1, \infty] \) with \( \frac{1}{p} + \frac{1}{q} = 1 \). Prove that
\[
| Tr (AB) | \leq \| A \|_{S^p} \| B \|_{S^q}.
\]
(Here, \(\| A \|_{S^p} \) is the \(p\)-Schatten norm of \( A \), defined by
\[
\| A \|_{S^p} = \left( \sum_{i=1}^n |\lambda_i|^p \right)^{1/p},
\]
where \( \lambda_1, \lambda_2, \dots, \lambda_n \) are the eigenvalues of \( A \).)

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Solution: 2015-22 An integral

Evaluate the following integral \[ \int_0^\infty \frac{\cos cx}{(\cosh x)^2} \, dx\] for a real constant \(c\).

The best solution was submitted by Sunghyuk Park (박성혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-22.

Alternative solutions were submitted by 신준형 (2015학번, +3), 이종원 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3), 유찬진 (2015학번, +2), 최인혁 (2015학번, +2), 이예찬 (오송고등학교 교사, +2), Luis F. Abanto-Leon (+2).

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2015-23 Fixed points

Let \(f:[0,1)\to[0,1)\)  be a function such that \[ f(x)=\begin{cases} 2x,&\text{if }0\le 2x\lt 1,\\ 2x-1, & \text{if } 1\le 2x\lt 2.\end{cases}\] Find all \(x\) such that \[ f(f(f(f(f(f(f(x)))))))=x.\]

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Solution: 2015-21 Differentiable function

Assume that a function \( f : (0, 1) \to [0, \infty) \) satisfies \( f(x) = 0 \) at all but countably many points \( x_1, x_2, \cdots \). Let \( y_n = f(x_n) \). Prove that, if \( \sum_{n=1}^{\infty} y_n < \infty \), then \( f \) is differentiable at some point.

The best solution was submitted by Jang, Kijoung (장기정, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-21.

Alternative solutions were submitted by 박성혁 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3), 최인혁 (2015학번, +3), 신준형 (2015학번, +2).

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