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).