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),

Show that for any triangle T and any Jordan curve C in the Euclidean plane, there exists a triangle inscribed in C which is similar to T.

The best solution was submitted by 강한필 (전산학부 2016학번, +4). Congratulations!

Here is his solution of problem 2021-02.

Other solutions was submitted by 하석민 (수리과학과 2017학번, +3), 박은아 (수리과학과 2015학번, +2).

Prove that for any given positive integer \( n \), there exists a sequence of the following operations that transforms \( n \) to a single-digit number (in decimal representation).

1) multiply a given positive integer by any positive integer.

2) remove all zeros in the decimal representation of a given positive integer.

The best solution was submitted by 강한필 (전산학부 2016학번, +4). Congratulations!

Here is his solution of problem 2021-01.

Other solutions was submitted by 김기수 (수리과학과 2018학번), 박은아 (수리과학과 2015학번, +3), 전해구 (기계공학과 졸업생).

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).

Let \( S \) be the unit sphere in \( \mathbb{R}^n \), centered at the origin, and \( P_1 P_2 \dots P_{n+1} \) a regular simplex inscribed in \( S \). Prove that for a point \( P \) inside \( S \),
\[
\sum_{i=1}^{n+1} (PP_i)^4
\]
depends only on the distance \( OP \) (and \(n\)).

The best solution was submitted by 이준호 (수리과학과 2016학번, +4). Congratulations!

Here is his solution of problem 2020-22.

Other solutions was submitted by 고성훈 (수리과학과 2018학번, +3), 채지석 (수리과학과 2016학번, +3).

Consider the cards with labels \( 1,\dots, n \) in some order. If the top card has label \(m \), we reverse the order of the top \( m \) cards. The process stops only when the card with label \( 1\) is on the top. Prove that the process must stop in at most \( (1.7)^n \) steps.

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

Here is his solution of problem 2020-18.

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

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.