SCHWEITZER 2007 Problem 3

Denote by $\omega (n)$ the number of prime divisors of the natural number $n$ (without multiplicities). Let $$F(x)=\max_{n\leq x} \omega (n) \,\,\,\,\,\,\,\,\,\,\,\, G(x)=\max_{n\leq x} \left( \omega (n) + \omega (n^2+1)\right)$$ Prove that $G(x)-F(x)\to \infty$ as $x\to\infty$ .

(translated by Miklós Maróti)