Evaluate the following integral \[ \int_0^\infty \frac{\cos cx}{(\cosh x)^2} \, dx\] for a real constant \(c\).
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Evaluate the following integral \[ \int_0^\infty \frac{\cos cx}{(\cosh x)^2} \, dx\] for a real constant \(c\).
Prove or disprove the following statement:
There exists a function \( f : \mathbb{R} \to \mathbb{R} \) such that
(1) \( f \equiv 0 \) almost everywhere, and
(2) for any nonempty open interval \(I\), \( f(I) = \mathbb{R} \).
The best solution was submitted by Joonhyung Shin (신준형, 2015학번). Congratulations!
Here is his solution of problem 2015-20.
Alternative solutions were submitted by 박성혁 (수리과학과 2014학번, +3, his solution), 이영민 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3, his solution), 장기정 (수리과학과 2014학번, +3, his solution), 최인혁 (2015학번, +3), 김동률 (2015학번, +2), 이신영 (물리학과 2012학번, +2), 송교범 (서대전고등학교 2학년, +3).
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.
Evaluate \[ \sum_{k=0}^{n} 2^k \tan \frac{\pi}{4\cdot 2^{n-k}}.\]
The best solution was submitted by Jang, Kijoung (장기정, 수리과학과 2014학번). Congratulations!
Here is his solution of problem 2015-19.
Alternative solutions were submitted by 김기택 (2015학번, +3), 박성혁 (수리과학과 2014학번, +3, his solution), 박훈민 (수리과학과 2013학번, +3), 신준형 (2015학번, +3), 이상민 (수리과학과 2014학번, +3), 이영민 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 최인혁 (2015학번, +3), Luis F. Abanto-Leon (+3), 김강식 (포항공대 수학과 2013학번, +3), 엄태강 (포항공대 수학과 2014학번, +3), 임준휘 (포항공대 수학과 2014학번, +3).
Prove or disprove the following statement:
There exists a function \( f : \mathbb{R} \to \mathbb{R} \) such that
(1) \( f \equiv 0 \) almost everywhere, and
(2) for any nonempty open interval \(I\), \( f(I) = \mathbb{R} \).
What is the determinant of the \(n\times n\) matrix \(A_n=(a_{ij}) \) where \[ a_{ij}=\begin{cases} 1 ,&\text{if } i=j, \\ x, &\text{if }|i-j|=1, \\ 0, &\text{otherwise,}\end{cases}\] for a real number \(x\)?
The best solution was submitted by Shin, Joonhyung (신준형, 2015학번). Congratulations!
Here is his soluton of problem 2015-18.
Alternative solutions were submitted by 김동률 (2015학번, +3), 박지민 (전산학부 석사 2015학번, +3), 최인혁 (2015학번, +3), 박성혁 (수리과학과 2014학번, +2), 박훈민 (수리과학과 2013학번, +2), 이상민 (수리과학과 2014학번, +2), 이영민 (수리과학과 2012학번, +2), 이종원 (수리과학과 2014학번, +2), 이호일 (수리과학과 2013학번, +2), 장기정 (수리과학과 2014학번, +2), 함도규 (2015학번, +2).
Evaluate \[ \sum_{k=0}^{n} 2^k \tan \frac{\pi}{4\cdot 2^{n-k}}.\]
The problem of the week will take a break during the midterm exam period and return on October 30, Friday. Good luck on your midterm exams!
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 \( i, j = 1, 2, \cdots, N-1 \).
The best solution was submitted by Park, Hun Min (박훈민, 수리과학과 2013학번). Congratulations!
Here is his solution of problem_2015_17.
Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 최인혁 (2015학번, +3), 박성혁 (수리과학과 2014학번, +3, solution), 신준형 (2015학번, +3), 이영민 (수리과학과 2012학번, +3), 장기정 (수리과학과 2014학번, +3), 함도규 (2015학번, +3).
What is the determinant of the \(n\times n\) matrix \(A_n=(a_{ij}) \) where \[ a_{ij}=\begin{cases} 1 ,&\text{if } i=j, \\ x, &\text{if }|i-j|=1, \\ 0, &\text{otherwise,}\end{cases}\] for a real number \(x\)?