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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 $n=k+1$.

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    $\begingroup$ A permutation such that $\sum \frac{P_i}{Q_i}$ .... what? $\endgroup$ – David Peterson Dec 21 '14 at 11:49
  • $\begingroup$ flagged "unclear what you are asking" $\endgroup$ – Dheeraj Kumar Dec 21 '14 at 11:51
  • $\begingroup$ "Is it possible to find a permutation such that sum(Pi/Qi)"... this is half of a question... $\endgroup$ – 5xum Dec 21 '14 at 11:52
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The geometric mean of those $n$ ratios is $1$. Therefore their arithmetic mean is at least $1$. Therefore their sum is at least $n$.

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  • $\begingroup$ Thanks a lot for your guide $\endgroup$ – Meh Dec 21 '14 at 12:31
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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.

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  • $\begingroup$ I want to use Induction algorithm and I prove that for n=2 the statement is incorrect but I cannot continue it could you help me? $\endgroup$ – Meh Dec 21 '14 at 12:12
  • $\begingroup$ @Meh If the statement is untrue for $n=2$, then the statement "this statement is true for all $n$" is false. No induction needed. $\endgroup$ – 5xum Dec 21 '14 at 12:54

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