be a triangle and 
its incenter. The lines 
and 
intersect the circumcircle of 
,
and 
be the circumferences of diameters 
and 
,
at 
at 
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 samrocksnature, Adventure10, Mango247
Let segments 
be tangent to circle 
at points 
, respectively. 
is a point on the extension of the segment 
and 
is the other meet point of the circle and the circumcircle of 
. 
is on 
such that 
is perpendicular to . Prove that
![\[ \angle MPQ=2\angle KML. \]](//latex.artofproblemsolving.com/2/0/e/20e2dc52aa1942c238b102e4023ada3705781017.png)
be tangent to circle 
at points 
, respectively. 
is a point on the extension of the segment 
and 
is the other meet point of the circle and the circumcircle of 
. 
is on 
such that 
is perpendicular to . Prove that![\[ \angle MPQ=2\angle KML. \]](http://latex.artofproblemsolving.com/2/0/e/20e2dc52aa1942c238b102e4023ada3705781017.png)
This post has been edited 1 time. Last edited by djmathman, Jun 23, 2015, 8:34 PM
Reason: latex
Reason: latex
and 
is triangle. Thus 
passes through the mid point of arc 
passes through the mid point of ar
be the reflection of point 
and 
is same.
