ÖMO 2020 Problem 3

Let $a$ be a fixed positive integer and $(e_n)$ the sequence, which is defined by $e_0=1$ and $$e_n=a + \prod_{k=0}^{n-1} e_k$$ for $n \geq 1$ .

Prove that (a) There exist infinitely many prime numbers that divide one element of the sequence. (b) There exists one prime number that does not divide an element of the sequence.

(Theresia Eisenkölbl)