In a graduation ceremony, \(n\) graduating students form a circle and their diplomas are distributed uniformly at random. Students who have their own diploma leave, and each of the remaining students passes the diploma she has to the student on her right, and this is one round. Again, each student with her own diploma leave and each of the remaining students passes the diploma to the student on her right and repeat this until everyone leaves. What is the probability that this process takes exactly \(k \) rounds until everyone leaves.

The best solution was submitted by 고성훈 (수리과학과 2018학번, +4). Congratulations!

Here is the best solution of problem 2021-12.

Let \(\mathcal{A}_n\) be the collection of all sequences \( \mathbf{a}= (a_1,\dots, a_n) \) with \(a_i \in [i]\) for all \(i\in [n]=\{1,2,\dots, n\}\). A nondecreasing \(k\)-subsequence of \(\mathbf{a}\) is a subsequence \( (a_{i_1}, a_{i_2},\dots, a_{i_k}) \) such that \(i_1< i_2< \dots < i_k\) and \(a_{i_1}\leq a_{i_2}\leq \dots \leq a_{i_k}\). For given \(k\), determine the smallest \(n\) such that any sequence \(\mathbf{a}\in \mathcal{A}_n\) has a nondecreasing \(k\)-subsequence.

The best solution was submitted by 고성훈 (수리과학과 2018학번, +4). Congratulations!

Here is his solution of problem 2021-06.

Another solution was submitted by 강한필 (전산학부 2016학번, +3).

For an \( n \times n \) matrix \( M \) with real eigenvalues, let \( \lambda(M) \) be the largest eigenvalue of \( M\). Prove that for any positive integer \( r \) and positive semidefinite matrices \( A, B \),

\[[\lambda(A^m B^m)]^{1/m} \leq [\lambda(A^{m+1} B^{m+1})]^{1/(m+1)}.\]

The best solution was submitted by 고성훈 (수리과학과 2018학번, +4). Congratulations!

Here is his solution of problem 2021-04.

Another solutions was submitted by 김건우 (수리과학과 2017학번, +3),

For each \( i \in \mathbb{N}\), let \(F_i\) be the \(i\)-th Fibonacci number where \(F_0=0, F_1=1\) and \(F_{i+1}=F_{i}+F_{i-1}\) for each \(i\geq 1\).
For \(n>m\), we divide \(F_n\) by \(F_m\) to obtain the remainder \(R\). Prove that either \(R\) or \(F_m-R\) is a Fibonacci number.

The best solution was submitted by 고성훈 (수리과학과 2018학번, +4). Congratulations!

Here is his solution of problem 2020-24.

Other solutions was submitted by Abdirakhman Ismail (2020학번), 이준호 (수리과학과 2016학번, +3).

Either find an example of a group which is expressed as the union of two proper subgroups or prove that such a group cannot exist.

The best solution was submitted by 고성훈 (수리과학과 2018학번). Congratulations!

Here is his solution of problem 2020-02.

Other solutions were submitted by 구은한 (수리과학과 2019학번, +3), 김기수 (수리과학과 2018학번, +3), 김동률 (수리과학과 2015학번, +3), 박현영 (전기및전자공학부 2016학번, +3), 유찬진 (수리과학과 2015학번, +3), 이준호 (2016학번, +3), 장우영 (서울대 경제학과, +3).

Let \( A \) be an \( m \times n \) matrix and \( \delta \in (0, 1) \). Suppose that \( \| A^T A – I \| \leq \delta \). Prove that all singular values of \( A \) are contained in the interval \( (1-\delta, 1+\delta) \).

The best solution was submitted by 고성훈 (수리과학과 2018학번). Congratulations!

Here is his solution of problem 2019-21.

A similar solution was submitted by 김태균 (수리과학과 2016학번, +3). Incomplete solutions was submitted by 박재원 (2019학번, +2), 하석민 (수리과학과 2017학번, +2).