Solution: 2016-3 Non-finitely generated subgroup

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).

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2016-4 Distances in a tree

Let $$T$$ be a tree on $$n$$ vertices $$V=\{1,2,\ldots,n\}$$. For two vertices $$i$$ and $$j$$, let $$d_{ij}$$ be the distance between $$i$$ and $$j$$, that is the number of edges in the unique path from $$i$$ to $$j$$. Let $$D_T(x)=(x^{d_{ij}})_{i,j\in V}$$ be the $$n\times n$$ matrix. Prove that $\det (D_T(x))=(1-x^2)^{n-1}.$

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Solution: 2016-2 Integral limit

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.

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Solution: 2016-1 Flipping Signs

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.

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2016-3 Non-finitely generated subgroup

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.

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2016-1 Flipping Signs: remains open

There were 13 submissions for problem 2016-1 but no correct solutions were submitted so far.

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2016-2 Integral limit

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

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2016-1 Flipping signs

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

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