,![\[
\frac{ab}{b+1} + \frac{bc}{c+1} + \frac{ca}{a+1} \geq \frac{3abc}{1 + abc}
\]](http://latex.artofproblemsolving.com/d/5/8/d582a3269c92041834c02153ee8e946a274e52bd.png)
and 
be positive integers.
and 
.
.
![\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
\]](http://latex.artofproblemsolving.com/3/2/b/32b41635c159921e11245369e66185ec69b0d785.png)
Y by Adventure10, Mango247
The quadrilateral 
has 
parallel to 
. 
is on the side and 
on the side such that 
. M is the intersection of 
and 
, and 
is the intersection of 
and 
. Find 
in terms of and .
has 
parallel to 
. 
is on the side and 
on the side such that 
. M is the intersection of 
and 
, and 
is the intersection of 
and 
. Find 
in terms of and .
and 
(both by A.A similitude criterion) (*). Hence
.
.
!
