[/quote]
,
and 
Prove that
,
touch sides 
at 
,
be the intersections of 
with 
,
be the projections of 
onto line 
be the second intersections of 
with the incircle 
and 
.
bisects segment 
.
be the incentre of 
such that the incircle touches the sides 
and 
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 
having 
.
is the leading coefficient of 
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
.
.
that satisfy 
are positive real numbers, then
Prove that
such that 
is a perfect square for all integers 
Prove that
and 
Prove that
satisfied 
Find minimum and maximum of 
Y by Adventure10, Mango247
Let 
be a cyclic quadrilateral. Let 
be the intersection of 
and 
. Line 
intersects the circumcircle of triangle 
in points 
and 
, with between 
and 
. Line 
intersects the circumcircle of triangle 
in points 
and 
, with between 
and 
. Prove that 
= 
= 
= 
.
be a cyclic quadrilateral. Let 
be the intersection of 
and 
. Line 
intersects the circumcircle of triangle 
in points 
and 
, with between 
and 
. Line 
intersects the circumcircle of triangle 
in points 
and 
, with between 
and 
. Prove that 
= 
= 
= 
.
.
so 
so 
then 
so 
.
.
the power of 
and 
are equal, so 
so 
is cyclic.
and 
but because 
.
