Open-problem statements, with their sources attached.

Let $a_1 < a_2 < \dots$ be a sequence of integers such that $\lim_{n\to\infty} \frac{a_n}{a_{n-1}^2} = 1$ and $\sum \frac{1}{a_n} \in \mathbb{Q}$.

Then, for all sufficiently large $n \ge 1$, $a_n = a_{n-1}^2 - a_{n-1} + 1$.

The Flint Hills series summing $csc(n)^2 / n^3$ from $n=1$ to $\infty$ converges. (Note that we 0-index the series below.)

The Cookson Hills series summing $sec(n)^2 / n^3$ from $n=1$ to $\infty$ converges.