[/quote]
,
such that 
and has exactly 
are positive real numbers, 
then
such that:
with all 
is real number
opposite the vertex 
be tangent to the side 
at the point 
.
on 
and 
on 
and 
lies on the circumcircle of triangle 
be the circumcircle of triangle 
is the midpoint of 
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 
respectively. 
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 Adventure10
The quadrilateral 
is inscribed in a circle of radius 
, so that 
is a diameter of the circumference and 
. A variable point 
moves along the semicircle determined by that does not contain 
or 
. Determine the position of for which the sum of the distances from to lines 
and 
is maximum.
is inscribed in a circle of radius 
, so that 
is a diameter of the circumference and 
. A variable point 
moves along the semicircle determined by that does not contain 
or 
. Determine the position of for which the sum of the distances from to lines 
and 
is maximum.
