MathOverflow

I am a junior high school student interested in exponential Diophantine equations. Recently, I have been investigating the following equation with mixed bases and exponents:
$$x^z + y^y = z^x$$
For positive integer solutions $(x, y, z) \in \mathbb{Z}^+$, it is easy to find some trivial families if we allow $x=1$ (for instance, $(1, y, 1+y^y)$ is a solution for any $y$)....

A friend recently nerd-sniped me with a (seemingly elementary) geometry question:
Let $\triangle ABC$ (with corresponding side lengths $a$, $b$, and $c$) be an obtuse triangle with $\angle C > 90^\circ$. Let $h$ be the length perpendicular drawn from $A$ to $BC$. Suppose that $\triangle ABC$ has integer side lengths and $h/a ...

As you all know, the Collatz conjecture claims that any positive integer will eventrually be reduced to 1 by appllying the sequence $n_{i+1} = x*n_{i} + 1$, when $n_{i}$ is odd, and $n_{i+1} = n_{i} / 2$, when $n_i$ is even, where $x = 3$.
I decided to extend Collatz conjecture and consider the same function for any other $x$. It turned out that already at $x = 5$ almost all sequences formed by the number n went to infinity, that is, its terms rapidly increased....

I would like to construct a "rich structure" intersection-closed family of sets $\mathcal{U}_n$ that is universal for the class of intersection-closed family of sets $\mathcal{F}$ such that $\varnothing,\bigcup_{S\in\mathcal{F}}S\in\mathcal{F}$ and ...