Solution: 2017-10 An inequality for determinant

Let $$A$$, $$B$$ be matrices over the reals with $$n$$ rows. Let $$M=\begin{pmatrix}A &B\end{pmatrix}$$. Prove that $\det(M^TM)\le \det(A^TA)\det(B^TB).$

The best solution was submitted by Lee, Bonwoo (이본우, 17학번). Congratulations!

Here is his solution of problem 2017-10.

Alternative solutions were submitted by Huy Tung Nguyen (2016학번, +3), 조태혁 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +2). One incorrect solution was received.

GD Star Rating

2017-11 Infinite series

Find the value of
$\sum_{n=1}^{\infty} \frac{1+ \frac{1}{2} + \dots + \frac{1}{n}}{n(2n-1)}.$

GD Star Rating

Solution: 2017-09 A Diophantine Equation

Find all positive integers $$a, b, c$$ satisfying $3^a + 5^b = 2^c.$

The best solution was submitted by Huy Tùng Nguyễn (2016학번). Congratulations!

Here is his solution of problem 2017-09.

Alternative solutions were submitted by 조태혁 (수리과학과 2014학번, +3), 이본우 (2017학번, +3), 최대범 (수리과학과 2016학번, +2), 이재우 (함양고등학교 2학년, +2).

GD Star Rating

2017-10 An inequality for determinant

Let $$A$$, $$B$$ be matrices over the reals with $$n$$ rows. Let $$M=\begin{pmatrix}A &B\end{pmatrix}$$. Prove that $\det(M^TM)\le \det(A^TA)\det(B^TB).$

GD Star Rating

2017-09 A Diophantine Equation

Find all positive integers $$a, b, c$$ satisfying
$3^a + 5^b = 2^c.$

GD Star Rating

Solution: 2017-08 Long arithmetic progression

Does there exist a constant $$\varepsilon>0$$ such that for each positive integer $$n$$ and each subset $$A$$ of $$\{1,2,\ldots,n\}$$ with $$\lvert A\rvert<\varepsilon n$$, there exists an artihmetic progression $$S$$ in $$\{1,2,\ldots,n\}$$ such that $$S\cap A=\emptyset$$ and $$\lvert S\rvert >\varepsilon n$$?

The best solution was submitted by Huy Tùng Nguyễn (2016학번). Congratulations!

Here is his solution of problem 2017-8.

Alternative solutions were submitted by 조태혁 (수리과학과 2014학번, +3), 위성군 (수리과학과 2015학번, +3), 최인혁 (물리학과 2015학번, +3, solution), 오동우 (수리과학과 2015학번, +3), 최대범 (수리과학과 2016학번, +3), 이본우 (2017학번, +3), 김태균 (수리과학과 2016학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3), 이재우 (함양고등학교 2학년, +3), 장기정 (수리과학과 2014학번, +2).

GD Star Rating

Solution: 2017-07 Supremum of a series

For $$\theta>0$$, let
$f(\theta) = \sum_{n=1}^{\infty} \left( \frac{1}{n+ \theta} – \frac{1}{n+ 3\theta} \right).$
Find $$\sup_{\theta > 0} f(\theta)$$.

The best solution was submitted by Oh, Dong Woo (오동우, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2017-07.

Alternative solutions were submitted by 조태혁 (수리과학과 2014학번, +3), 위성군 (수리과학과 2015학번, +3), 최인혁 (물리학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), Huy Tung Nguyen (2016학번, +3), 이본우 (2017학번, +3), 김태균 (수리과학과 2016학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3).

GD Star Rating
Does there exist a constant $$\varepsilon>0$$ such that for each positive integer $$n$$ and each subset $$A$$ of $$\{1,2,\ldots,n\}$$ with $$\lvert A\rvert<\varepsilon n$$, there exists an artihmetic progression $$S$$ in $$\{1,2,\ldots,n\}$$ such that $$S\cap A=\emptyset$$ and $$\lvert S\rvert >\varepsilon n$$?