Let \( n \in \mathbb{Z}^+ \) and \( x, y \in \mathbb{R}^+ \) such that \( x^n + y^n = 1 \). Prove that
\[
(1-x)(1-y) \left( \sum_{k=1}^n \frac{1+x^{2k}}{1+x^{4k}} \right) \left( \sum_{k=1}^n \frac{1+y^{2k}}{1+y^{4k}} \right) < \frac{7}{10}. \]

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

Here is his solution of problem 2019-17.

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

Suppose a group \(G\) has a finite index subgroup that maps onto the free group of rank 2. Show that every countable group can be embedded in one of the quotient groups of \(G\).

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

Here is his solution of problem 2019-16.

Let \( A_{a, b} = \{ (x, y) \in \mathbb{Z}^2 : 1 \leq x \leq a, 1 \leq y \leq b \} \). Consider the following property, which we call Property R:

“If each of the points in \(A\) is colored red, blue, or yellow, then there is a rectangle whose sides are parallel to the axes and vertices have the same color.”

Find the maximum of \(|A_{a, b}|\) such that \( A_{a, b} \) has Property R but \( A_{a-1, b} \) and \( A_{a, b-1} \) do not.

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

Here is his solution of problem 2019-13.

An incorrect solution was received. Late solutions are not graded.

Two players play a game with a polynomial with undetermined coefficients
\[
1 + c_1 x + c_2 x^2 + \dots + c_7 x^7 + x^8.
\]
Players, in turn, assign a real number to an undetermined coefficient until all coefficients are determined. The first player wins if the polynomial has no real zeros, and the second player wins if the polynomial has at least one real zero. Find who has the winning strategy.

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

Here is his solution of problem 2018-23.

Alternative solutions were submitted by 채지석 (수리과학과 2016학번, +3), 권홍 (중앙대 물리학과, +2).

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.

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).