[/quote]
,
such that 
and has exactly 
are positive real numbers, 
then
such that:
with all 
is real number
opposite the vertex 
at the point 
.
on 
and 
on 
lies on the circumcircle of triangle 
be the circumcircle of triangle 
be the second intersection of 
and 
be a transversal of 
be intersections of 
with 
be a tangent line of 
is cyclic
be positive integers such that 
and 
.
.
and 
Prove that
at 
,
be the intersections of 
with 
,
be the projections of 
onto line 
be the second intersections of 
with the incircle 
be the intersection of 
and 
.
bisects segment 
.
be the incentre of 
such that the incircle touches the sides 
and 
meet the incircle at 
respectively. 
and 
intersect at 
Prove that 
bisects 
then prove 
![$P\in\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/f/3/d/f3d763f4197792a2956273485eb1e94700a43044.png)
has degree 
.
is the leading coefficient of 
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
.
.
Y by centslordm, Instance
Two spheres are externally tangent at point 
. The segments 
and 
touch the spheres with 
and 
lying on the first sphere and 
and 
on the second. Let 
and 
be the projections of the midpoints of segments 
and 
on the line connecting the centers of the spheres. Prove that 
.
. The segments 
and 
touch the spheres with 
and 
lying on the first sphere and 
and 
on the second. Let 
and 
be the projections of the midpoints of segments 
and 
on the line connecting the centers of the spheres. Prove that 
.
and let 
be midpoints of segments 
respectively; obviously points 
lie in a radical plane of spheres, so easy to see, that 
,
so we are done.
