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0<b_i-b_j<1+3\sqrt[3]{b_ib_j}

Source: Peru Cono TST 2020 P5

September 11, 2023
algebra

Problem Statement

Find the smallest positive integer nn such that for any nn distinct real numbers b1,b2,…,bnb_1, b_2,\ldots ,b_n in the interval [1,1000][ 1, 1000 ] there always exist bib_i and bjb_j such that: 0<bi−bj<1+3bibj30<b_i-b_j<1+3\sqrt[3]{b_ib_j}
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