Prove or disprove that a surjective homomorphism from a finitely generated abelian group to itself is an isomorphism.

The best solution was submitted by 김유일 (2020학번). Congratulations!

Here is his solution of problem 2020-17.

A different solution was submitted by 박은아 (수리과학과 2015학번, +3).

Let \( A \) be an \( n \times n \) Hermitian matrix and \( \lambda_1 (A) \geq \lambda_2 (A) \geq \dots \geq \lambda_n (A) \) the eigenvalues of \( A \). Prove that for any \( 1 \leq k \leq n \)
\[
A \mapsto \lambda_1 (A) + \lambda_2 (A) + \dots + \lambda_k (A)
\]
is a convex function.

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

Here is his solution of problem 2020-16.

Other solutions were submitted by 길현준 (수리과학과 2018학번, +3), 이준호 (수리과학과 2016학번, +3).

Let \( m_0=n \). For each \( i\geq 0 \), choose a number \( x_i \) in \( \{1,\dots, m_i\} \) uniformly at random and let \( m_{i+1}= m_i – x_i\). This gives a random vector \( \mathbf{x}=(x_1,x_2, \dots) \). For each \( 1\leq k\leq n\), let \( X_k \) be the number of occurrences of \( k \) in the vector \( \mathbf{x} \).

For each \(1\leq k\leq n\), let \(Y_k\) be the number of cycles of length \(k\) in a permutation of \( \{1,\dots, n\} \) chosen uniformly at random. Prove that \( X_k \) and \(Y_k\) have the same distribution.

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

Here is his solution of problem 2020-15.

Say there are n points. For each pair of points, we add an edge with probability 1/3. Let \(P_n\) be the probability of the resulting graph to be connected (meaning any two vertices can be joined by an edge path). What can you say about the limit of \(P_n\) as n tends to infinity?

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

Here is his solution of problem 2020-14.

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 김건우 (수리과학과 2017학번, +3), 이준호 (수리과학과 2016학번, +3), 김유일 (2020학번, +3).

There are \(n\) people participating to a chess tournament and every two players play exactly one game against each other. The winner receives \(1\) point and the loser gets \(0\) point and if the game is a draw, each player receives \(0.5\) points. Prove that if at least \(3/4\) of the games are draws, then there are two players with the same total scores.

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

Here is his solution of problem 2020-12.

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

Show that there is a subgroup of a free group of ran 2 that is not finitely generated.

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

Here is his solution of problem 2020-11.

Other solutions were submitted by 조한슬 (수리과학과 2017학번, +3), 최백규(생명과학과 대학원, +2).

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)