Find all positive integers $a, b, c$ satisfying $$3^a + 5^b = 2^c.$$

The best solution was submitted by Huy Tùng Nguyễn (2016학번). Congratulations!

Here is his solution of problem 2017-09.

Alternative solutions were submitted by 조태혁 (수리과학과 2014학번, +3), 이본우 (2017학번, +3), 최대범 (수리과학과 2016학번, +2), 이재우 (함양고등학교 2학년, +2).

Does there exist a constant $\varepsilon>0$ such that for each positive integer $n$ and each subset $A$ of $\{1,2,\ldots,n\}$ with $\lvert A\rvert<\varepsilon n$, there exists an artihmetic progression $S$ in $\{1,2,\ldots,n\}$ such that $S\cap A=\emptyset$ and $\lvert S\rvert >\varepsilon n$?

The best solution was submitted by Huy Tùng Nguyễn (2016학번). Congratulations!

Here is his solution of problem 2017-8.

Alternative solutions were submitted by 조태혁 (수리과학과 2014학번, +3), 위성군 (수리과학과 2015학번, +3), 최인혁 (물리학과 2015학번, +3, solution), 오동우 (수리과학과 2015학번, +3), 최대범 (수리과학과 2016학번, +3), 이본우 (2017학번, +3), 김태균 (수리과학과 2016학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3), 이재우 (함양고등학교 2학년, +3), 장기정 (수리과학과 2014학번, +2).

Suppose that $f : (2, \infty) \to (-2, 2)$ is a continuous function and there exists a positive constant $m$ such that $| 1 + xf(x) + (f(x))^2 | \leq m$ for any $x > 2$. Prove that, for any $x > 2$,
$$\left| f(x) – \frac{\sqrt{x^2 -4}-x}{2} \right| \leq 6 \sqrt{m}.$$

The best solution was submitted by Huy Tùng Nguyễn (2016학번). Congratulations!

Here is his solution of problem 2017-05.

Alternative solutions were submitted by 위성군 (수리과학과 2015학번, +3), 조태혁 (수리과학과 2014학번, +3), 최인혁 (물리학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 오동우 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 김재현 (수리과학과 2016학번, +3), 김태균 (수리과학과 2016학번, +2).