![\[
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0
\]](http://latex.artofproblemsolving.com/7/8/f/78f8e6c2570684631dc288144e13568d71e68ec8.png)
.
and 
.
such that 
are the lengths of a right triangle.
be reals such that 
. Prove that
Let 
. Prove that
.
are the midpoints of 
respectively. 
is the midpoint of arc 
,
is symmetric to 
.
be the incenter and tangent of angles 
respectively. Prove that 
have a common point
and 
.
Y by
Let 
be a scalene triangle, 
a point inside the side 
a point inside the side 
and 
a point inside the side 
.
a) If
determine if it is necessarily true that 
and are the midpoints of the sides of ABC.
b) If
, determine if it is necessarily true that and are the midpoints of the sides of .
be a scalene triangle, 
a point inside the side 
a point inside the side 
and 
a point inside the side 
.a) If
determine if it is necessarily true that 
and are the midpoints of the sides of ABC.b) If
, determine if it is necessarily true that and are the midpoints of the sides of .
