Permutation and inequality

Use AM - GM inequality for the $'n'$ fractions ,

$\dfrac{\dfrac{a_1}{b_1}+\dfrac{a_2}{b_2}+\dots +\dfrac{a_n}{b_n}}{n} \geq \sqrt[n]{\dfrac{a_1}{b_1}\times \dfrac{a_2}{b_2}\times \dots \times\dfrac{a_n}{b_n}}$

Since $b_1 , b_2 , b_3 \dots b_n$ is a permutation of $a_1 , a_2 , a_3 \dots a_n$ , $R.H.S$ becomes $1$ .

$\dfrac{\dfrac{a_1}{b_1}+\dfrac{a_2}{b_2}+\dots +\dfrac{a_n}{b_n}}{n} \geq \sqrt[n]{1}$

$\dfrac{a_1}{b_1}+\dfrac{a_2}{b_2}+\dots +\dfrac{a_n}{b_n} \geq n$

Since here $n=1234567890$ ,

$\displaystyle \sum_{r=1}^{1234567890} \dfrac{a_r}{b_r} \geq \large \boxed{1234567890}$