Category Archives: problem

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.

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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?

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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 \(i\neq j\).

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2025-03 Distinct sums under shifts

Consider any sequence \( a_1,\dots, a_n \) of non-negative integers in \(\{0,1,\dots, m\}\). Prove that \[|\{ a_i+ a_j + (j-i): 1\leq i < j \leq n \}|\geq m \] when \(m= \lfloor \frac{1}{4} n^{2/3} \rfloor \).

A bonus problem: Can you find a function \(f(n)=\omega(n^{2/3})\) such that the above statement is true when \(m = f(n) \)? Is there such a function with \(f(n)= \Omega(n)\)? (You would still get full points without answering the bonus question.)

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2024-19 Stationary function

Let \(g(t): [0,+\infty) \to [0,+\infty)\) be a decreasing continuous function. Assume \(g(0)=1\), and for every \(s, t \geq 0 \) \[t^{11}g(s+t) \leq 2024 \; [g(s)]^2.\] Show that \(g(11) = g(12)\).

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