Monthly Archives: April 2025

2025-06 Know thy hats!

There are \(n+1\) hats, each labeled with a number from \(1\) to \(n+1\), and \(n\) people. Each person is randomly assigned exactly one hat, and each hat is assigned to at most one person (i.e., the assignment is injective). A person can see all other assigned hats but cannot see their own hat and the unassigned hat. Each person must independently guess the number on their own hat.

If everyone correctly guesses their own hat’s number, they win; otherwise, they lose. They may discuss a strategy before the hats are assigned, but no communication is allowed afterward.

Determine a strategy that maximizes their probability of winning.

GD Star Rating
loading...

Solution: 2025-05 Commutativity and the matrix exponential

Let \( X \in \mathbb{R}^{n \times n} \) be a symmetric matrix with eigenvalues \( \lambda_i \) and orthonormal eigenvectors \( u_i \). The spectral decomposition gives \( X = \sum_{i=1}^n \lambda_i u_i u_i^\top \). For a function \( f : \mathbb{R} \to \mathbb{R} \), define \( f(X) := \sum_{i=1}^n f(\lambda_i) u_i u_i^\top \). Let \( X, Y \in \mathbb{R}^{n \times n} \) be symmetric. Is it always true that \( e^{X+Y} = e^X e^Y \)? If not, under what conditions does the equality hold?


The best solution was submitted by 이명규 (전기및전자공학부 20학번, +4). Congratulations!

Here is the best solution of problem 2025-05.

Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), 김준홍 (수리과학과 석박통합과정, +3), 신민규 (수리과학과 24학번, +3), 정서윤 (수리과학과 학사과정, +3), 채지석 (수리과학과 석박통합과정, +3).

GD Star Rating
loading...

2025-05 Commutativity and the matrix exponential

Let \( X \in \mathbb{R}^{n \times n} \) be a symmetric matrix with eigenvalues \( \lambda_i \) and orthonormal eigenvectors \( u_i \). The spectral decomposition gives \( X = \sum_{i=1}^n \lambda_i u_i u_i^\top \). For a function \( f : \mathbb{R} \to \mathbb{R} \), define \( f(X) := \sum_{i=1}^n f(\lambda_i) u_i u_i^\top \). Let \( X, Y \in \mathbb{R}^{n \times n} \) be symmetric. Is it always true that \( e^{X+Y} = e^X e^Y \)? If not, under what conditions does the equality hold?

GD Star Rating
loading...

Solution: 2025-04 Multivariate polynomials

We write \(tx = (tx_0,…,tx_5)\) for \(x=(x_0,…,x_5)\in \mathbb{R^{6}}\) and \(t\in \mathbb{R}\). Find all real multivariate polynomials \(P(x)\) in \(x\) satisfying the following properties:
(a) \(P(tx) = t^d P(x)\) for all \(t\in \mathbb{R}\) and \(x\in \mathbb{R}^{6}\), where \(0\leq d \leq 15\) is an integer;
(b) \(P(x) =0\) if \(x_i = x_j\) with \(i\neq j\).


The best solution was submitted by 채지석 (수리과학과 석박통합과정, +4). Congratulations!

Here is the best solution of problem 2025-04.

Other solutions were submitted by 이명규 (전기및전자공학부 20학번), 김동훈 (수리과학과 22학번, +3), 김준홍 (수리과학과 석박통합과정, +3), 신민규 (수리과학과 24학번, +3), 정서윤 (수리과학과 학사과정, +3), Anar Rzayev (수리과학과 19학번, +3).

GD Star Rating
loading...