MHB What is the minimum sum of fractions with positive numbers and permutations?

[lfdahl]

Let $a_1,a_2, ... , a_n$ be positive numbers.

Let $i_1,i_2, ... , i_n$ be a permutation of $1,2,...,n$.

Determine the smallest possible value of the sum:

$$\sum_{k=1}^{n}\frac{a_k}{a_{i_k}}$$

 

[Olinguito]

Spoiler

By AM–GM,
$$\sum_{k=1}^n\frac{a_k}{a_{i_k}}\ \ge\ n\cdot\sqrt[n]{\frac{a_1\cdots a_n}{a_{i_1}\cdots a_{i_n}}}\ =\ n.$$
This is attained when $i_k=k$ for $k=1,\ldots,n$ (i.e. when it’s the identity permutation).

Hence the minimum value is $n$.

 

[lfdahl]

Great job, Olinguito! Thankyou for your participation!(Handshake)