# The order of an automorphism

Let $A\in \mathrm{GL}(n, q)$. What is the maximum order $A$ can have?

# No point left behind

Every infinite set admits a metric with no isolated points.

# Walk like an Egyptian

Let $n>0$ be a fixed positive integer and let $a_1,\ldots, a_n$ be distinct positive integers such that

$\dfrac{1}{a_1}+\ldots+\dfrac{1}{a_n}<1.$

Find the maximum possible value of $\dfrac{1}{a_1}+\ldots+\dfrac{1}{a_n}$.

# Dense totients

Show that the set of numbers of the form $\varphi(m)/m$ is dense in the closed unit interval.

# Keeping it real

Let $A\in\Bbb{R}^{n\times n}$ be a matrix with a real, simple eigenvalue. Prove that there exists a neighborhood of $A$ in which all matrices have at least one real eigenvalue.

# Compact function spaces

Let $K$ be a compact metric space, $\varphi:K\to(0,\infty)$ a continuous function and $A=\{f\in C(K) : |f(x)|\leq\varphi(x)\}$. Prove that $A$ is compact iff $K$ is finite.