be a triangle and 
its incenter. The lines 
and 
intersect the circumcircle of 
and 
,
and 
be the circumferences of diameters 
and 
,
at 
and 
,
and 
.
.
,
and 
.
is the intersection point of the common external tangents of a random pair of these incircles and Y is the intersection point of common external tangents of some other pair of these three incircles, then 
passes through either 
,
or 
.
be positive real numbers such that : 
. Prove that : 
with![\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]](http://latex.artofproblemsolving.com/a/b/f/abf10d7e679f6f04db556424a0a0eda26f371fd4.png)
for all 
.
Y by
In the square 
, the points 
and 
belong to the sides 
and 
, respectively. The diagonal 
intersects 
and 
at points 
and 
, respectively, If 
, and 
, prove that the lines 
and 
are perpendicular.
, the points 
and 
belong to the sides 
and 
, respectively. The diagonal 
intersects 
and 
at points 
and 
, respectively, If 
, and 
, prove that the lines 
and 
are perpendicular.This post has been edited 1 time. Last edited by parmenides51, Mar 18, 2021, 5:44 AM
