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 \(a_1\), \(a_2\), \(\ldots\), \(a_m\) 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).

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 \(A_1,A_2,A_3,\ldots,A_n\) 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).

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).

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).

Find all integers \( n \) such that \( \sqrt{1} + \sqrt{2} + \dots + \sqrt{n} \) is an integer.

The best solution was submitted by Han, Junho (한준호, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2018-03.

Alternative solutions were submitted by 김태균 (수리과학과 2016학번, +3), 이본우 (수리과학과 2017학번, +3), 이종원 (수리과학과 2014학번, +3, solution), 채지석 (수리과학과 2016학번, +3), 최백규 (2016학번, +3), 최인혁 (물리학과 2015학번, +3), 김건우 (수리과학과 2017학번, +2). Two incorrect solutions were received.

For \(n\ge 1\), let \(f(x)=x^n+\sum_{k=0}^{n-1} a_k x^k \) be a polynomial with real coefficients. Prove that if \(f(x)>0\) for all \(x\in [-2,2]\), then \(f(x)\ge 4\) for some \(x\in [-2,2]\).

The best solution was submitted by Choi, Inhyeok (최인혁, 물리학과 2015학번). Congratulations!

Here is his solution of problem 2018-02.

Alternative solutions were submitted by 이본우 (수리과학과 2017학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이종원 (수리과학과 2014학번, +3), 채지석 (수리과학과 2016학번, +3), 하석민 (수리과학과 2017학번, +3), 한준호 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +2), 이재우 (함양고등학교 3학년, +2).

Define a sequence \( \{ a_n \} \) by \( a_1 = a \) and
\[
a_n = \frac{2n-1}{n-1} a_{n-1} -1
\]
for \( n \geq 2 \). Find all real values of \( a \) such that \( \lim_{n \to \infty} a_n \) exists.

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

Here is his solution of problem 2018-01.

Alternative solutions were submitted by 강한필 (전산학부 2016학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이종원 (수리과학과 2014학번, +3), 채지석 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), 한준호 (수리과학과 2015학번, +3), 고성훈 (2018학번, +2), 김태균 (수리과학과 2016학번, +2), 송교범 (고려대 수학과 2017학번, +2), 이재우 (함양고등학교 3학년, +2), 노우진 (물리학과 2015학번) 및 윤정인 (물리학과 2016학번) (+2). Two incorrect solutions were received.