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.
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.
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.
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).
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?
POW 2020-13 is still open and anyone who first submits a correct solution will get the full credit.
Let \( a_n \) be a sequence defined recursively by \( a_0 = a_1 = \dots = a_5 = 1 \) and
\[
a_n = \frac{a_{n-1} a_{n-5} + a_{n-2} a_{n-4} + a_{n-3}^2}{a_{n-6}}
\]
for \( n \geq 6 \). Prove or disprove that every \( a_n \) is an integer.