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inequality, asymmetric, nonhomogeneous, with a 7 and a radical!

Source: IMO Shortlist 2018 A7, by Evan Chen, Taiwan

July 17, 2019
IMO Shortlist

Problem Statement

Find the maximal value of S=ab+73+bc+73+cd+73+da+73,S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}}, where aa, bb, cc, dd are nonnegative real numbers which satisfy a+b+c+d=100a+b+c+d = 100.
Proposed by Evan Chen, Taiwan
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