RMM 2011 Problem 4

Given a positive integer $n=\prod_{i=1}^{s} p_{i}^{\alpha_{i}}$, we write $\Omega(n)$ for the total number $\sum_{i=1}^{s} \alpha_{i}$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n)=(-1)^{\Omega(n)}$ (so, for example, $\left.\lambda(12)=\lambda\left(2^{2} \cdot 3^{1}\right)=(-1)^{2+1}=-1\right)$.

Prove the following two claims: i) There are infinitely many positive integers $n$ such that $\lambda(n)=\lambda(n+1)=+1 ;$ ii) There are infinitely many positive integers $n$ such that $\lambda(n)=\lambda(n+1)=-1$. (ROMANIA) DAN SchwARZ