Find the smallest positive integer $n$ such that for any $n$ distinct real numbers $b_1, b_2,\ldots ,b_n$ in the interval $[ 1, 1000 ]$ there always exist $b_i$ and $b_j$ such that: $$0<b_i-b_j<1+3\sqrt[3]{b_ib_j}$$
Source: Peru Cono TST 2020 P5
Tags: algebra
Find the smallest positive integer $n$ such that for any $n$ distinct real numbers $b_1, b_2,\ldots ,b_n$ in the interval $[ 1, 1000 ]$ there always exist $b_i$ and $b_j$ such that: $$0<b_i-b_j<1+3\sqrt[3]{b_ib_j}$$