Let \(A\) be a \(2\times 2\) matrix. Prove that if \(Av_1=\lambda_1v_1\) and \(Av_2=\lambda_2v_2\) for distinct reals \(\lambda_1\) and \(\lambda_2\) and nonzero vectors \(v_1\) and \(v_2\), then both columns of \(A-\lambda_1 I\) is a multiple of \(v_2\).

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

Here is his solution of problem 2018-12.

Alternative solutions were submitted by 고성훈 (2018학번, +3), 권홍 (중앙대 물리학과, +3), 길현준 (2018학번, +3), 김태균 (수리과학과 2016학번, +3), 이본우 (수리과학과 2017학번, +3), 채지석 (수리과학과 2016학번, +3), 한준호 (수리과학과 2015학번, +3).

For a positive integer \( n \), let \( S(n) \) be the sum of all decimal digits in \( n \), i.e., if \( n = n_1 n_2 \dots n_m \) is the decimal expansion of \( n \), then \( S(n) = n_1 + n_2 + \dots + n_m \). Find all positive integers \( n \) and \( r \) such that \( (S(n))^r = S(n^r) \).

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

Here is his solution of problem 2018-09.

Alternative solutions were submitted by 채지석 (수리과학과 2016학번, +3), 한준호 (수리과학과 2015학번, +3), 이본우 (수리과학과 2017학번, +3), 권홍 (중앙대 물리학과, +2).

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).

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 \(x_1x_2+x_2x_3+x_3x_4+\cdots + x_{n-1}x_n+x_nx_1\)?

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

Here is his solution of problem 2018-04.

Alternative solutions were submitted by 이본우 (수리과학과 2017학번, +3), 채지석 (수리과학과 2016학번, +3), 고성훈 (2018학번, +2).

A rich old man had \( k \) sons and \( N \) camels in the herd. The will of the father stated that his \( r \)-th son should receive \( 1/ N_r\) of his camels for \( r = 1, 2, \dots, k\). Since \( N+1 \) is a common multiple of \( N_1, N_2, \dots, N_k \), the sons could not divided \( N \) camels as their father wished. The sons visited a wise man to solve the issue. The wise man listened about the will, and he brought his own camel, which he added to the herd. The herd was then divided up according to the old man’s wishes. The wise man then took back the one camel that remained, which was his own. For given \( k \), find the maximal number of camels \( N \equiv N(k) \) for which there is a solution to the problem where \( N_1, N_2, \dots, N_k\) are positive integers.

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

Here is his solution of problem 2016-13.

Alternative solutions were submitted by 최인혁 (물리학과 2015학번, +3), 국윤범 (수리과학과 2015학번, +3), 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (2016학번, +3), 김재현 (2016학번, +2), 김태균 (2016학번, +2), 한준호 (수리과학과 2015학번, +2), 이상민 (수리과학과 2014학번, +2). One incorrect solution was submitted.

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.