Mathematical Reflections 2012, Issue 5 - Problem J241

Problem:
Determine all positive integers that can be represented as $$\dfrac{ab+bc+ca}{a+b+c+\min (a,b,c)}$$ for some positive integers $a,b,c$.

Proposed by Titu Andreescu.

Solution:

We claim that every positive integer $n$ can be represented in this form. Suppose $a \geq b \geq c$. Let $c=1, b=n$. Then
$$\dfrac{ab+bc+ca}{a+b+c+\min (a,b,c)}=\dfrac{a(n+1)+n}{a+n+2}=n+1-\dfrac{n^2+2n+2}{a+n+2},$$ so it suffices to choose $a=n^2+n$.