Normal
\[S_1:=\left(4a+\frac{1}{a}\left)+\left(4b+\frac{1}{b}\left)+\left(4c+\frac{1}{c}\right)-3(a+b+c)\rightarrow DONE!\]\[S_2+8= (a+b+c+d) \left(\frac{1}{b+c+d}+\frac{1}{a+c+d}+\frac{1}{a+b+d}+\frac{1}{a+b+c}\right)+(a+b+c+d) \left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right) \rightarrow DONE!\rightarrow \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d} \ge \frac{16}{a+b+c+d}\]\[\left(a^2+\frac{1}{b^2}\right)(1+16):=\left(a+ \frac{4}{b} \right)^2+\left(4a-\frac{1}{b} \right)^2\rightarrow DONE!!\]
\[S_1:=\left(4a+\frac{1}{a}\left)+\left(4b+\frac{1}{b}\left)+\left(4c+\frac{1}{c}\right)-3(a+b+c)\rightarrow DONE!\]
\[S_2+8= (a+b+c+d) \left(\frac{1}{b+c+d}+\frac{1}{a+c+d}+\frac{1}{a+b+d}+\frac{1}{a+b+c}\right)+(a+b+c+d) \left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right) \rightarrow DONE!\rightarrow \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d} \ge \frac{16}{a+b+c+d}\]
\[\left(a^2+\frac{1}{b^2}\right)(1+16):=\left(a+ \frac{4}{b} \right)^2+\left(4a-\frac{1}{b} \right)^2\rightarrow DONE!!\]
