Solution: 2015-19 Sum of tangent functions

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).

GD Star Rating
loading...

2015-20 Dense function

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 \(I\), \( f(I) = \mathbb{R} \).

GD Star Rating
loading...

Solution: 2015-18 Determinant

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).

GD Star Rating
loading...

Midterm break

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!

GD Star Rating
loading...

Solution: 2015-17 Inverse of a minor

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).

GD Star Rating
loading...

2015-18 Determinant

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 \(x\)?

GD Star Rating
loading...

Solution: 2015-16 Complex integral

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).

GD Star Rating
loading...

2015-17 Inverse of a minor

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 \( i, j = 1, 2, \cdots, N-1 \).

GD Star Rating
loading...

Solution: 2015-15 A sequence periodic modulo m for all m

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).

GD Star Rating
loading...