[/quote]
,
be positive integers such that 
and 
.
.
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 
be the intersection of 
and 
.
bisects segment 
.
be the incentre of 
such that the incircle touches the sides 
and 
respectively. 
and 
meet the incircle at 
respectively. 
and 
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
Y by Adventure10
Let 
be a triangle and 
a point on the side 
. Let 
be the circumference with center 
and radius 
that cuts the side 
at 
such that lies between 
and . Let 
be the circumference with center 
and radius 
that cuts the side 
at 
such that lies between and . Line 
cuts and at 
and 
different from , respectively. We call 
the point of intersection of 
and 
. Prove that 
be a triangle and 
a point on the side 
. Let 
be the circumference with center 
and radius 
that cuts the side 
at 
such that lies between 
and . Let 
be the circumference with center 
and radius 
that cuts the side 
at 
such that lies between and . Line 
cuts and at 
and 
different from , respectively. We call 
the point of intersection of 
and 
. Prove that 
= 
and 
,
+ 
,
= 
so 
= 
