Permutation of positive real numbers

Question

Consider a set of positive real numbers $\{P_1,P_2,\dots,P_n\}$ and a permutation of this set $\{Q_1,Q_2,\dots,Q_n\}$.

Is it possible to find a permutation such that $$\sum_{i=1}^n\frac{P_i}{Q_i}<n$$?

We can prove that for $n=2$ the statement is incorrect and assume that for $n=k$ the statement is correct and we should prove for $n=k+1$.

Answer 1

The geometric mean of those $n$ ratios is $1$. Therefore their arithmetic mean is at least $1$. Therefore their sum is at least $n$.

Answer 2

If you can prove that the statement is incorrect for $n=2$, then that is all you need to do. The statement is false, period.