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Problem 8.3

Part of 2022 Bulgarian Spring Math Competition

Problems(1)

Prove or disprove inequalities

Source: 2022 Bulgarian Spring Math Competition, Problem 8.3

3/27/2022
Given the inequalities: a)a) (2ab+c)2+(2ba+c)2+(2ca+b)2≥ac+ba+cb\left(\frac{2a}{b+c}\right)^2+\left(\frac{2b}{a+c}\right)^2+\left(\frac{2c}{a+b}\right)^2\geq \frac{a}{c}+\frac{b}{a}+\frac{c}{b} b)b) (a+bc)2+(b+ca)2+(c+ab)2≥ab+bc+ca+9\left(\frac{a+b}{c}\right)^2+\left(\frac{b+c}{a}\right)^2+\left(\frac{c+a}{b}\right)^2\geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+9 For each of them either prove that it holds for all positive real numbers aa, bb, cc or present a counterexample (a,b,c)(a,b,c) which doesn't satisfy the inequality.
algebraInequalityinequalities