Evaluate the following integral for \( z \in \mathbb{C}^+ \).\[\frac{1}{2\pi} \int_{-2}^2 \log (z-x) \sqrt{4-x^2} dx.\]
The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!
Here is his solution of problem 2015-16.
Alternative solutions were submitted by 최인혁 (2015학번, +2), 박훈민 (수리과학과 2013학번, +2), 박성혁/이경훈 (수리과학과 2014학번, +2).
Let \( H \) be an \( N \times N \) positive definite matrix and \( G = H^{-1} \). Let \( H’ \) be an \( (N-1) \times (N-1) \) matrix obtained by removing the \( N \)-th row and the column of \( H \), i.e., \( H’_{ij} = H_{ij} \) for any \( i, j = 1, 2, \cdots, N-1 \). Let \( G’ = (H’)^{-1} \). Prove that
\[
G_{ij} – G’_{ij} = \frac{G_{iN} G_{Nj}}{G_{NN}}
\]
for any \( i, j = 1, 2, \cdots, N-1 \).
Does there exist an infinite sequence such that (i) every integer appears infinitely many times and (ii) the sequence is periodic modulo \(m\) for every positive integer \(m\)?
The best solution was submitted by Park, Sunghyuk (박성혁, 수리과학과 2014학번). Congratulations!
Here is his solution of problem 2015-15.
Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 신준형 (2015학번, +3), 최인혁 (2015학번, +2), 이영민 (수리과학과 2012학번, +2), 장기정 (수리과학과 2014학번, +2).
Evaluate the following integral for \( z \in \mathbb{C}^+ \).
\[
\frac{1}{2\pi} \int_{-2}^2 \log (z-x) \sqrt{4-x^2} dx.
\]
Find all positive integers \(n\) such that the following statement holds:
Let \(f:\mathbb{R}^n\to \mathbb {R}\) be a differentiable function that has a unique critical point \(c\). If \(f\) has a local maximum at \(c\), then \(f(c)\) is an absolute maximum of \(f\).
The best solution was submitted by Choi, Inhyeok (최인혁, 2015학번). Congratulations!
Here is his solution of problem 2015-14.
Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 김재준 (2014학번, +3), 박훈민 (수리과학과 2013학번, +3), 박성혁 (수리과학과 2014학번, +3), 오동우 (2015학번, +3), 이영민 (수리과학과 2012학번, +3), 장기정 (수리과학과 2014학번, +3), 신준형 (2015학번, +2). One incorrect solutions were received (LAL). Delayed submissions were not graded.
Does there exist an infinite sequence such that (i) every integer appears infinitely many times and (ii) the sequence is periodic modulo \(m\) for every positive integer \(m\)?
Find all positive integers \(n\) such that the following statement holds:
Let \(f:\mathbb{R}^n\to \mathbb {R}\) be a differentiable function that has a unique critical point \(c\). If \(f\) has a local maximum at \(c\), then \(f(c)\) is an absolute maximum of \(f\).
Find the minimum value of
\[
\int_{\mathbb{R}} f(x) \log f(x) dx
\]
among functions \(f: \mathbb{R} \rightarrow \mathbb{R}^+ \cup \{ 0 \}\) that satisfy the condition
\[
\int_{\mathbb{R}} f(x) dx = \int_{\mathbb{R}} x^2 f(x) dx = 1.
\]
The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!
Here is his solution of problem 2015-13.
Alternative solutions were submitted by 김경석 (2015학번, +3), 김재준 (2014학번, +3), 김희주 (2015학번, +2), 박성혁 (수리과학과 2014학번, +2), 박훈민 (수리과학과 2013학번, +3), 신준형 (2015학번, +3), 오동우 (2015학번, +2), 이신영 (물리학과 2012학번, +2), 이영민 (수리과학과 2012학번, +2), 이정환 (2015학번, +3), 장기정 (수리과학과 2014학번, +2), 최인혁 (2015학번, +2), Luis F. Abanto-Leon (+2), 이시우 (포항공대 수학과 2013학번, +3). Two incorrect solutions (L.S.M., H.I.S.) were submitted.
Find the minimum value of
\[
\int_{\mathbb{R}} f(x) \log f(x) dx
\]
among functions \(f: \mathbb{R} \rightarrow \mathbb{R}^+ \cup \{ 0 \}\) that satisfy the condition
\[
\int_{\mathbb{R}} f(x) dx = \int_{\mathbb{R}} x^2 f(x) dx = 1.
\]
이수철 (장려상), 이종원 (최우수상), 이창옥 교수 (학과장), 김기현 (우수상), 엄태현 (우수상), 엄상일 교수
Thanks all for participating POW actively. Here’s the list of winners:
이종원 (수리과학과 2014학번) 38
김기현 (수리과학과 2012학번) 37
진우영 (수리과학과 2012학번) 37
엄태현 (수리과학과 2012학번) 37
이수철 (수리과학과 2012학번) 36
고경훈 (2015학번) 27
오동우 (2015학번) 23
정성진 (수리과학과 2013학번) 21
최인혁 (2015학번) 21
이명재 (수리과학과 2012학번) 18
이영민 (수리과학과 2012학번) 18
함도규 (2015학번) 18
김경석 (2015학번) 15
장기정 (수리과학과 2014학번) 12
박훈민 (수리과학과 2013학번) 9
최두성 (수리과학과 2011학번) 7
유찬진 (2015학번) 6
국윤범 (2015학번) 5
박성혁 (수리과학과 2014학번) 5
이상민 (수리과학과 2014학번) 5
김기택 (2015학번) 4
김동률 (2015학번) 3
김동철 (수리과학과 2013학번) 3
신준형 (2015학번) 3
윤준기 (수리과학과 2014학번) 3
이병학 (수리과학과 2013학번) 3
홍혁표 (수리과학과 2013학번) 3
Muhammadfiruz Hassnov (2014학번) 3
윤지훈 (수리과학과 2012학번) 2