[/quote]
,
be positive integers such that 
and 
.
.
and 
Prove that
touch sides 
at 
,
perpendicular to 
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 
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
Y by
Let the points 
and 
be the intersections of the inscribed circle of the right-angled triangle 
, with sides 
and 
respectively , and points 
and 
respectively be the intersections of the ex-scribed circles opposite to vertices 
and 
with direction 
. Prove that the quadrilateral 
is a cyclic if and only if the triangle is right-angled with a right angle at the vertex 
.
and 
be the intersections of the inscribed circle of the right-angled triangle 
, with sides 
and 
respectively , and points 
and 
respectively be the intersections of the ex-scribed circles opposite to vertices 
and 
with direction 
. Prove that the quadrilateral 
is a cyclic if and only if the triangle is right-angled with a right angle at the vertex 
.
.
where 
is the 
.
.
.
and thus 
and we are done.
