with 
.![\[
(x^{2}-yz)(y^{2}-zx)(z^{2}-xy).
\]](http://latex.artofproblemsolving.com/a/8/7/a87fae27584012a867bc0a90190c0317f9f7ab07.png)
be a positive integer. A positive integer 
is called a benefactor of 
and 
such that 
.
is a benefactor of 
because 
.
benefactors.
and 
: Prove that :
Y by
Let 
be a triangle, 
and 
the feet of its altitudes from 
and 
, respectively, 
its touchpoints of the incircle with the sides 
and 
, respectively. The circumcircles of triangles 
and 
intersect again at 
. The circumcircles of triangles 
and 
intersect again at 
. The circumcircles of triangles 
and 
intersect again at 
. Prove that the points 
are collinear.
Malik Talbi
be a triangle, 
and 
the feet of its altitudes from 
and 
, respectively, 
its touchpoints of the incircle with the sides 
and 
, respectively. The circumcircles of triangles 
and 
intersect again at 
. The circumcircles of triangles 
and 
intersect again at 
. The circumcircles of triangles 
and 
intersect again at 
. Prove that the points 
are collinear.Malik Talbi
and 
be the orthocenter and incenter of 
and 
.
.
and 
are both perpendicular to 
,
and it suffices to show 
.
.
are collinear.
