I4.3 Let $x, y$ and $z$ be positive integers. If $|x-y|^{2010}+|z-x|^{2011}=b$ and $c=|x-y|+|y-z|+|z-x|$, find the value of $c$.
Reference: 1996 FI2.3, 2005FI4.1, 2006 FI4.2, 2013 FI1.4, 2015 HG4, 2015 FI1.1
Clearly $|x-y|$ and $|z-x|$ are non-negative integers $|x-y|^{2010}+|z-x|^{2011}=1 \Rightarrow(|x-y|=0$ and $|z-x|=1)$ or $(|x-y|=1$ and $|z-x|=0)$ When $x=y$ and $|z-x|=1, c=0+|y-z|+|z-x|=2|z-x|=2$
When $|x-y|=1$ and $|z-x|=0, c=1+|y-z|+0=1+|y-x|=1+1=2$