Isn't this IMO 2008?

Denote $$m=\frac1{1-p},~n=\frac1{1-q},~s=\frac1{1-r}.$$Then $mns=\frac1{\prod(1-p)}$, while $$(1-m)(1-n)(1-s)=\prod\frac p{1-p}=\frac1{\prod(1-p)}.$$The latter used $pqr=1$. From $mns=(1-m)(1-n)(1-s)$ expand to get $$m+n+s=1+mn+ns+sm.$$Let $mn+ns+sm=x$ then $m+n+s=x+1$. So the quantity on the left side of the desired inequality is \begin{align*}&m^2+n^2+s^2=(m+n+s)^2-2(mn+ns+sm)\\=&(x+1)^2-2x=x^2+1\ge1.\end{align*}