Middle European Mathematical Olympiad 2011 Problem T-8

We call a positive integer $n$ amazing if there exist positive integers $a,b,c$ such that the equality $$n=(b,c)(a,bc)+(c,a)(b,ca)+(a,b)(c,ab)$$ holds. Prove that there exist $2011$ consecutive positive integers which are amazing.

(By $(m,n)$ we denote the greatest common divisor of positive integers $m$ and $n$.)