Elegant but pugnacious inequality

If $p=30$, $q=r=5$, and $s=18$, $t=u=1$, then the condition is satisfied, but $p+q+r-2(s+t+u)=0$. Furthermore, for sufficiently small and positive $\epsilon$,

let $p=30$, $q=5$, $r=5$,

and $s=18-6\epsilon$, $t=1+\epsilon$, $u=1$.

Then $\frac{s}{p}+\frac{t}{q}+\frac{u}{r}=1$.

However $$\frac{p+q+r}{2}-s-t-u=5\epsilon,$$ thus

$$\frac{-3}{\frac{p+q+r}{2}-s-t-u} = -\frac{3}{5\epsilon}.$$

If positive $\epsilon$ approaches to $0$, then the above term goes to $-\infty$.

It makes a contradiction.