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A strange inequality in Taiwan

Source: Own

August 2, 2024
inequalitiescalculus

Problem Statement

Given positive real a,b,ca,b,c satisfying 1a+1+3b+3+3c+3=72\frac{1}{\sqrt{a+1}}+\frac{3}{\sqrt{b+3}}+\frac{3}{\sqrt{c+3}}=\frac72 Prove that abc≤3abc\leq 3.\\ I was asked to propose a inequality for the condition of abc<3abc<3 inequality since <3 looks like a heart shape, then I construct a equality and with the help of wolfram, I gave the birth of this bad-looking inequality, I’m glad to see any method besides calculus.
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