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).

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).

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).

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).

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).

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).

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).

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).

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).

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.