Recently, I have found this problem:
We call a positive integer $n$ amazing if there exists 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 exists $2011$ consecutive positive integers which are amazing.
Here some amazing numbers:
In the picture from left to right you find the numbers: $n$, $a$, $b$ and $c$.
I have tried to solve this problem in a lot of different ways, for example using the definition of $GCD$, or divisibility but I can't go on. Any idea?
Note:by $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.