During high school I participated in a few math contests, and in undergrad I wrote several problems for the Caltech Harvey Mudd Math Competition and USA Math Talent Search. Every two weeks I will post a recreational (non-research) math problem which I encountered during this time (and particularly enjoyed) below.

Fibonacci Binomials

The Fibonacci sequence $(F_n)$ is defined by setting $F_0 = F_1 = 1$, and then taking $$F_{n} = F_{n-1} + F_{n-2}$$ for all integers $n\ge 2$. We further define the sequence of Fibonacci factorials $(f_n)$ by taking $$f_n = \prod_{m=0}^n F_m$$ for each integer $n\ge 0$. Prove that for all nonnegative integers $a$ and $b$, the quantity $$\frac{f_{a+b}}{f_af_b}$$ is an integer.

A new problem will be posted here on August 12th, 2022.

Previously posted problems can be found here.