Let \( G \) be a subgroup of \( GL_2 (\mathbb{R}) \) generated by \( \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} \) and \( \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \). Let \( H \) be a subset of \( G \) that consists of all matrices in \( G \) whose diagonal entries are \( 1 \). Prove that \( H \) is a subgroup of \( G \) but not finitely generated.

The best solution was submitted by Jo, Tae Hyouk (조태혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2016-3.

Alternative solutions were submitted by 강한필 (2016학번, +3), 국윤범 (수리과학과 2015학번, +3), 김경석 (연세대학교 의예과 2016학번, +3), 김기택 (수리과학과 2015학번, +3), 김동규 (수리과학과 2015학번, +3), 김동률 (수리과학과 2015학번, +3), 송교범 (서대전고등학교 3학년, +3), 어수강 (서울대학교 수학교육과 박사과정, +3), 유찬진 (수리과학과 2015학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이정환 (수리과학과 2015학번, +3), 이종원 (수리과학과 2014학번, +3), 이준호 (2016학번, +3), 이태영 (2013학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (2016학번, +3), Muhammaadfiruz Hasanov (2014학번, +3), 배형진 (마포고등학교 2학년, +2), 이상민 (수리과학과 2014학번, +2).

For \( a \geq 0 \), find
\[
\lim_{n \to \infty} n \int_{-1}^0 \left( x + \frac{x^2}{2} + e^{ax} \right)^n dx.
\]

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

Here is his solution of problem 2016-02.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김동규 (수리과학과 2015학번, +3), 김동하 (기계공학과 2014학번, +3), 이상민 (수리과학과 2014학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이종원 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 최대범 (2016학번, +3), 최인혁 (물리학과 2015학번, +3), Muhammaadfiruz Hasanov (2014학번, +3), 이준호 (2016학번, +2). One incorrect solution was submitted.

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 \(k=1,2,\ldots,n-1\), \[ \left\lvert \sum_{i=1}^k \varepsilon_i x_i-\sum_{i=k+1}^n \varepsilon_i x_i \right\rvert\le 1.\]

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2016-1.

Alternative solutions were submitted by 노희광 (화학과 2014학번, +2), 안현수 (2016학번, +2), 이상민 (수리과학과 2014학번, +2), 홍혁표 (수리과학과 2013학번, +2). There were 10 incorrect submissions.

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).

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).