For a permutation \(\pi: [n]\rightarrow [n]\), we define the displacement of \(\pi\) to be \(\sum_{i\in [n]} |i-\pi(i)|\).

For given \(k\), prove that the number of even permutations of \([n]\) with displacement \(2k\) minus the number of odd permutations of \([n]\) with displacement \(2k\) is \((-1)^{k}\binom{n-1}{k}\).

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

Here is his solution of problem 2020-09.

Another solution was submitted by 고성훈 (수리과학과 2018학번, +3).

In the problem 2019-08 (https://mathsci.kaist.ac.kr/pow/2019/2019-08-group-action/), we considered a group G acting by isometries on a proper geodesic metric space X properly discontinuously and cocompactly. Such an action is called a geometric action. The conclusion was that a geometric action leads to that G is finitely generated.

Would this conclusion still hold in the case the space X is not necessarily proper?

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

Here is his solution of problem 2020-08.

Suppose that \( x, y, z \) are positive integers satisfying
\[
0 \leq x^2 + y^2 – xyz \leq z+1.
\]
Prove that \( x^2 + y^2 – xyz \) is a perfect square.

The best solution was submitted by 임상호 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-07.

Another solution was submitted by 김기수 (수리과학과 2018학번, +3), 홍의천 (수리과학과 2017학번, +3)