Question:
Determine the smallest possible dimension of matrices $M_i$ with entries in either $\mathbb{C}$ or $\mathbb{Z}/m \mathbb{Z}$ such that:

1. For $1 \le i \le n, M_i^2=0$.
2. Every product of the permutations of $\{M_i\}$ does not vanish: $\prod_{i} M_i \ne 0$.
3. Matrices $M_i$ commute with each other.

Answer: 2^n


Question: Find the limit of the expression \( (1+x)^{\frac{y}{x}} \) as \((x, y)\) approaches \((0, 2)\).

Answer: e^2


Question: Determine whether the condition $g \ge p+1$ is necessary and sufficient for the existence of a $g \times (gp)$ matrix arrangement, where each column is a permutation of $\{1, \dots, g\}$, each number in $\{1, \dots, g\}$ appears $p$ times in each row, and any two columns agree in at most one entry.

Answer: g \ge p + 1


Question: Determine the number of distinct prime ideals in the ring \( \mathbb{Q}[x]/(x^4-1) \).

Answer: 3


Question: Determine the largest possible value of $k \in \mathbb{N}$ such that there exists an integer $n$ for which the digit sum of $k$ consecutive perfect squares, starting from $(n+1)^2$ to $(n+k)^2$, is a perfect square for each square in the sequence. Note that $k \geq 7$, and an example of $k=8$ has been found: $S(46045846^2) = 8^2, S(46045847^2) = 7^2, \ldots, S(46045853^2) = 8^2$. What is this largest $k$?

Answer: 8


Question: Determine if the congruence $n \equiv 1 \pmod{\lambda^5}$, where $\lambda = 1 - \zeta_5$ and $\zeta_5$ is a 5th root of unity, can be simplified to a congruence modulo an integer in $\mathbb{Z}$.

Answer: 25


Question: Evaluate the integral \( \int_{0}^{\pi} \int_{\frac{x}{2}}^{x}\frac{\sin(x)}{x} dydx \).

Answer: 1


Question: Find the degree of the field extension \( \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n}) \) over \( \mathbb{Q} \), where \( p_1, p_2, \ldots, p_n \) are distinct primes.

Answer: 2^n


Question: Evaluate the surface integral \( \iint_{S} \text{curl}(\vec{F}) \cdot d\vec{S} \) where \( \vec{F}(x,y,z) = (y^2z, xz, x^2y^2) \) and \( S \) is the part of the paraboloid \( z = x^2 + y^2 \) that is inside the cylinder \( x^2 + y^2 = 1 \), with an outward facing normal.

Answer: \pi


Question: For a real symmetric matrix with repeated eigenvalues, is it always possible to find an orthogonal matrix \( P \) such that the eigen decomposition satisfies a given equation?

Answer: Yes


Question: Find a constant \( \beta_K < 1 \) such that for the function \( g(x) = \frac{1}{3}(x+\sqrt{x^2+4}) \), the inequality \( |g(x)-g(y)| \leq \beta_K |x-y| \) holds for all \( x, y \) in a compact subset \( K \) of \( \mathbb{R} \).

Answer: \dfrac{2}{3}