Solution: 2017-14 Polynomials of degree at most n

Let \(f(x)\in \mathbb R[x]\) be a polynomial of degree at most \(n\) such that \[ x^2+f(x)^2\le 1\] for all \( -1\le x\le 1 \). Prove that \( \lvert f'(x)\rvert \le 2(n-1)\) for all \( -1\le x\le 1\).

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

Here is his solution of problem 2017-14.

Alternative solutions were submitted by 유찬진 (수리과학과 2015학번, +3), 이본우 (2017학번, +2). One incorrect solution was submitted.

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Solution: 2017-13 Infinite series with recurrence relation

Let \(a_0 = a_1 =1\) and \(a_n = n a_{n-1} + (n-1) a_{n-2}\) for \(n \geq 2\). Find the value of
\[
\sum_{n=0}^{\infty} (-1)^n \frac{n!}{a_n a_{n+1}}.
\]

The best solution was submitted by Choi, Daebeom (최대범, 수리과학과 2016학번). Congratulations!

Here is his solution of problem 2017-13.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김동률 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 유찬진 (수리과학과 2015학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이본우 (2017학번, +3), 이태영 (수리과학과 2013학번, +3), 장기정 (수리과학과 2014학번, +3, solution), 조태혁 (수리과학과 2014학번, +3, solution), 최인혁 (물리학과 2015학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 김기택 (수리과학과 2015학번, +2), 이재우 (함양고등학교 2학년, +2), 정의현 (수리과학과 2015학번, +2).

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Solution: 2017-12 Invertible matrices

Let \(A\) and \(B\) be \(n\times n\) matrices. Prove that if \(n\) is odd and both \(A+A^T\) and \(B+B^T\) are invertible, then \(AB\neq 0\).

The best solution was submitted by Shin, Joonhyung (신준형, 수리과학과 2015학번). Congratulations!

Here is his solution of the problem 2017-12.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김동률 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 위성군 (수리과학과 2015학번, +3), 유찬진 (수리과학과 2015학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이본우 (2017학번, +3), 이준협 (하나고등학교, +3), 이태영 (수리과학과 2013학번, +3), 이형진 (청주대 수학교육과 2011학번, +3), 임성혁 (수리과학과 2016학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), Saba Dzmanashvili (2017학번, +3).

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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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