Does there exist a (possibly \(n\)-dependent) constant \( C \) such that
\[
\frac{C}{a_n} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2 \leq \frac{a_1+ \dots + a_n}{n} – \sqrt[n]{a_1 \dots a_n} \leq \frac{C}{a_1} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2
\]
for any \( 0 < a_1 \leq a_2 \leq \dots \leq a_n \)?

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

Here is his solution of problem 2018-21.

Alternative solutions were submitted by 하석민 (수리과학과 2017학번, +3),
이본우 (수리과학과 2017학번, +2). One incorrect submission was received.

Let \(f:\mathbb R\to\mathbb R\) be a function such that \[ -1\le f(x+y)-f(x)-f(y)\le 1\] for all reals \(x\), \(y\). Does there exist a constant \(c\) such that \( \lvert f(x)-cx\rvert \le 1\) for all reals \(x\)?

The best solution was submitted by Ha, Seokmin (하석민, 수리과학과 2017학번). Congratulations!

Here is his solution of problem 2018-20.

An alternative solution was submitted by 채지석 (수리과학과 2016학번, +3). There were two incorrect submissions.

Let
\[
f(x) = 1 + \left( \frac{1}{2} \cdot x \right)^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \cdot x^2 \right)^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot x^3 \right)^2 + \dots
\]
Prove that
\[
(\sin x) f(\sin x) f'(\cos x) + (\cos x) f(\cos x) f'(\sin x) = \frac{2}{\pi \sin x \cos x}.
\]

The best solution was submitted by Seo, Juneyoung (서준영, 수리과학과 대학원생). Congratulations!

Here is his solution of problem 2018-19.

Alternative solutions were submitted by 길현준 (2018학번, +3, solution), 김기현 (수리과학과 대학원생, +3), 이본우 (수리과학과 2017학번, +3).

Suppose that we are given 12 points evenly spaced on a circle. Starting from a point in the 12 o’clock position, a particle P will move to one of the adjacent positions with equal probably, 1/2. P stops if it visits all 12 points. What is the most likely point that P stops for the last?

The best solution was submitted by Ha, Seokmin (하석민, 수리과학과 2017학번). Congratulations!

Here is his solution of problem 2018-18.

An alternative solution was submitted by 채지석 (수리과학과 2016학번, +3).

For \( a > b > 0 \), find the value of
\[
\int_0^{\infty} \frac{e^{ax} – e^{bx}}{x(e^{ax}+1)(e^{bx}+1)} dx.
\]

The best solution was submitted by Ha, Seokmin (하석민, 수리과학과 2017학번). Congratulations!

Here is his solution of problem 2018-17.

Alternative solutions were submitted by 길현준 (2018학번, +3), 김태균 (수리과학과 2016학번, +3, solution), 이본우 (수리과학과 2017학번, +3), 채지석 (수리과학과 2016학번, +3), 서준영 (수리과학과 대학원생, +3), 이재우 (함양고등학교 3학년, +3).