# Permutation of positive real numbers

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 $n=k+1$.

• A permutation such that $\sum \frac{P_i}{Q_i}$ .... what? – David Peterson Dec 21 '14 at 11:49
• flagged "unclear what you are asking" – Dheeraj Kumar Dec 21 '14 at 11:51
• "Is it possible to find a permutation such that sum(Pi/Qi)"... this is half of a question... – 5xum Dec 21 '14 at 11:52

The geometric mean of those $n$ ratios is $1$. Therefore their arithmetic mean is at least $1$. Therefore their sum is at least $n$.
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.
• @Meh If the statement is untrue for $n=2$, then the statement "this statement is true for all $n$" is false. No induction needed. – 5xum Dec 21 '14 at 12:54