ABCD

2AB=BC

2CD=DE

2EF=FA

\frac{FA}{FC}+\frac{BC}{BE}+\frac{DE}{DA}\geqslant2.

AC=x

AE=y

CE=z

ABCE

ACDE

ACEF

AB(z+2y)=AB\cdot z+2AB\cdot y=AB\cdot z+BC\cdot y\geqslant BE\cdot x,

CD(y+2x)=CD\cdot y+2CD\cdot x=CD\cdot y+DE\cdot x\geqslant DA\cdot z,

EF(x+2z)=EF\cdot x+2EF\cdot z=EF\cdot x+FA\cdot z\geqslant FC\cdot y.

\frac{FA}{FC}+\frac{BC}{BE}+\frac{DE}{DA}=2\left(\frac{EF}{FC}+\frac{AB}{BE}+\frac{CD}{DA}\right)\geqslant\frac{2y}{x+2z}+\frac{2x}{z+2y}+\frac{2z}{y+2x}.

9u=x+2z,~9v=y+2x,~9w=z+2y.

x=4v-2w+u,~y=4w-2u+v,~z=4u-2v+w.

\frac{2y}{x+2z}+\frac{2x}{z+2y}+\frac{2z}{y+2x}=\frac{8w-4u+2}{9u}+\frac{8v-4w+2u}{9w}+\frac{8u-4v+2w}{9v}=

=\frac{2}{3}\left(\frac{u}{v}+\frac{v}{w}+\frac{w}{u}\right)+\frac{2}{9}\left(\frac{u}{v}+\frac{v}{u}\right)+\frac{2}{9}\left(\frac{v}{w}+\frac{w}{v}+\frac{w}{u}\right)+\frac{2}{9}\left(\frac{w}{u}+\frac{u}{w}+\frac{w}{u}\right)-\frac{4}{3}~\geqslant

\geqslant2\sqrt[{3}]{{\frac{u}{v}\cdot\frac{v}{w}\cdot\frac{w}{u}}}+\frac{4}{9}\sqrt{\frac{u}{v}\cdot\frac{v}{u}}+\frac{4}{9}\sqrt{\frac{v}{w}\cdot\frac{w}{v}}+\frac{4}{9}\sqrt{\frac{w}{u}\cdot\frac{u}{v}}-\frac{4}{3}=

=2+\frac{4}{9}+\frac{4}{9}+\frac{4}{9}-\frac{4}{3}=2.