# Solution: 2018-08 Large LCM

Let $$a_1$$, $$a_2$$, $$\ldots$$, $$a_m$$ be distinct positive integers. Prove that if $$m>2\sqrt{N}$$, then there exist $$i$$, $$j$$ such that the least common multiple of $$a_i$$ and $$a_j$$ is greater than $$N$$.

The best solution was submitted by Bonwoo Lee (이본우, 수리과학과 2017학번). Congratulations!

Here is his solution of problem 2018-08.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 김태균 (수리과학과 2016학번, +3), 한준호 (수리과학과 2015학번, +3), 이재우 (함양고등학교 3학년, +3).

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# Solution: 2018-07 A tridiagonal matrix

Let $$S$$ be an $$(n+1) \times (n+1)$$ matrix defined by
$S_{ij} = \begin{cases} (n+1)-i & \text{ if } j=i+1, \\ i-1 & \text{ if } j=i-1, \\ 0 & \text{ otherwise. } \end{cases}$
Find all eigenvalues of $$S$$.

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

Here is his solution of problem 2018-07.

Alternative solutions were submitted by 한준호 (수리과학과 2015학번, +3), 채지석 (수리과학과 2016학번, +3), Hitesh Kumar (Imperial College London, +2), 고성훈 (2018학번, +2).

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