# 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 $$S$$.

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# Solution: 2018-06 Product of diagonals

Let $$A_1,A_2,A_3,\ldots,A_n$$ be the vertices of a regular $$n$$-gon on the unit circle. Evaluate $$\prod_{i=2}^n A_1A_i$$. (Here, $$A_1A_i$$ denotes the length of the line segment.)

The best solution was submitted by Taegyun Kim (김태균, 수리과학과 2016학번). Congratulations!

Here is his solution of problem 2018-06.

Alternative solutions were submitted by 권홍 (중앙대 물리학과, +3), 고성훈 (2018학번, +3), 김건우 (수리과학과 2017학번, +3), 이본우 (수리과학과 2017학번, +3), 이종원 (수리과학과 2014학번, +3), 채지석 (수리과학과 2016학번, +3), 최백규 (생명과학과 2016학번, +3), 하석민 (수리과학과 2017학번, +3), 한준호 (수리과학과 2015학번, +3).

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# Solution: 2018-05 Roulette

A gambler is playing roulette and betting $1 on black each time. The probability of winning$1 is 18/38, and the probability of losing $1 is 20/38. Find the probability that starting with$20 the player reaches \$40 before losing the money.

The best solution was submitted by Jiseok Chae (채지석, 수리과학과 2016학번). Congratulations!

Here is his solution of problem 2018-05.

Alternative solutions were submitted by 고성훈 (2018학번, +3), 김건우 (수리과학과 2017학번, +3), 이종원 (수리과학과 2014학번, +3), 한준호 (수리과학과 2015학번, +3), 문정욱 (2018학번, +3), 이현우 (전산학부 대학원생, +3), 임동현 (전산학부 대학원생, +3), 이본우 (수리과학과 2017학번, +2).

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# 2018-06 Product of diagonals

Let $$A_1,A_2,A_3,\ldots,A_n$$ be the vertices of a regular $$n$$-gon on the unit circle. Evaluate $$\prod_{i=2}^n A_1A_i$$. (Here, $$A_1A_i$$ denotes the length of the line segment.)

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Let $$x_1,x_2,\ldots,x_n$$ be reals such that $$x_1+x_2+\cdots+x_n=n$$ and $$x_1^2+x_2^2+\cdots +x_n^2=n+1$$. What is the maximum of $$x_1x_2+x_2x_3+x_3x_4+\cdots + x_{n-1}x_n+x_nx_1$$?