MathNet 2022-part-02 — 197 problems (EN–VI) Question 141

Determine all positive integers $a$, $b$, $c$ for which there exist positive integers $x$, $y$, $z$, such that: $ab + 1 = x!$, $bc + 1 = y!$, $ca + 1 = z!$, where $n!$ denotes the product $1 \cdot 2 \cdot 3 \ldots \cdot n$.