CMO 2014 Problem 4

Let $n=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ be the prime factorisation of $n$ . Define $\omega(n)=t$ and $\Omega(n)=a_1+a_2+\ldots+a_t$ . Prove or disprove: For any fixed positive integer $k$ and positive reals $\alpha,\beta$ , there exists a positive integer $n>1$ such that i) $\frac{\omega(n+k)}{\omega(n)}>\alpha$ ii) $\frac{\Omega(n+k)}{\Omega(n)}<\beta$ .