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.

Prove or disprove the following statement: There exists a function whose Maclaurin series converges at only one point.

The best solution was submitted by Kook, Yun Bum (국윤범, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2017-21.

Alternative solutions were submitted by 민찬홍 (중앙대학교사범대학부속고등학교 3학년, +3), 채지석 (수리과학과 2016학번, +3), 최대범 (수리과학과 2016학번, +3), 하석민 (2017학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3). Four incorrect solutions were submitted, mostly due to misunderstanding.

Determine whether or not the following infinite series converges. \[ \sum_{n=0}^{\infty} \frac{ 1 }{2^{2n}} \binom{2n}{n}.\]

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

Here is his solution of problem 2017-20.

Alternative solutions were submitted by 고성훈 (+3), 국윤범 (수리과학과 2015학번, +3), 길현준 (인천과학고등학교 2학년, +3), 김태균 (수리과학과 2016학번, +3), 민찬홍 (중앙대학교사범대학부속고등학교 3학년, +3), 유찬진 (수리과학과 2015학번, +3), 이원웅 (건국대 수학과 2014학번, +3), 이준협 (하나고등학교, +3), 채지석 (2016학번, +3), 최대범 (수리과학과 2016학번, +3), 하석민 (2017학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 이준성 (상문고등학교 1학년, +3), 정경훈 (서울대학교 컴퓨터공학과, +3), Mirali Ahmadili & Saba Dzmanashvili (2017학번, +3).

For an integer \( p \), define
\[
f_p(n) = \sum_{k=1}^n k^p.
\]
Prove that
\[
\frac{1}{2} \sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_3(n)} + 2\sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_1(n)} = \sum_{n=1}^{\infty} \frac{(f_{-1}(n))^2}{f_1(n)}.
\]

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

Here is his solution of problem 2017-19.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 길현준 (인천과학고등학교 2학년, +3), 민찬홍 (중앙대학교사범대학부속고등학교 3학년, +3), 유찬진 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 채지석 (2016학번, +3), 최대범 (수리과학과 2016학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 이재우 (함양고등학교 2학년, +2), 하석민 (2017학번, +2), Saba Dzmanashvili & Mirali Ahmadili  (2017학번, +2).