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![$P\in\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/f/3/d/f3d763f4197792a2956273485eb1e94700a43044.png)
has degree 
having real roots in 
. Suppose 
is the leading coefficient of 
. Show that, 
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both and 
. Integers and are coprime if 
.
that satisfy 
are positive real numbers, then
be the center of circumcircle 
of acuted-triangle 
. Another circle 
with center 
is tangent internally to circle at point 
and side
at point 
. Also circle intersect the lines 
and 
respectively at points 
and 
, and lines 
and 
respectively at points 
and , 
and . Lines 
and 
intersect at
. Prove 
.
