I3. A.M.-G.M. Inequality
\[\prod_{cyc}(1+\frac{a}{b})=2+\sum_{sym}\frac{a}{b}=2+\frac{1}{3}\sum_{cyc}(\frac{a}{b}+\frac{a}{b}+\frac{b}{c})+\frac{1}{3}\sum_ {cyc}(\frac{c}{b}+\frac{a}{c}+\frac{a}{c})\geqslant 2+2\sum_{cyc}\sqrt[3]{\frac{a^2}{bc}}=2(1+\frac{a+b+c}{\sqrt[3]{abc}})\]
I4. Cauchy-Schwarz Inequality
\[\sum_{cyc}\frac{a}{2b+3c}=\sum_{cyc}\frac{a^2}{2ab+3ca}\geqslant \frac{(a+b+c)^2}{5ab+5bc+5ca}\geqslant \frac{3ab+3bc+3ca}{5ab+5bc+5ca}=\frac{3}{5}\]