such that:
and 
: Prove that :
,
are all polynomials such that ![\[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\]](http://latex.artofproblemsolving.com/1/6/a/16a71abc110ece558d427a07d7be2d1f72148024.png)
prove that 
is a factor of 
.
Y by Adventure10
Let 
be an acute-angled triangle and 
be a circle with 
as diameter intersecting 
and 
at 
and 
respectively. Tangents are drawn at 
and 
to intersect at 
. Show that the ratio of the circumcentre of triangle to that if 
is a rational number.
be an acute-angled triangle and 
be a circle with 
as diameter intersecting 
and 
at 
and 
respectively. Tangents are drawn at 
and 
to intersect at 
. Show that the ratio of the circumcentre of triangle to that if 
is a rational number.This post has been edited 1 time. Last edited by Rushil, Oct 16, 2005, 5:00 AM
.
of the circumradius of triangle ABC (this is a trivial property of the nine-point circle); in other words, the ratio between the circumradii of triangles ABC and YXR is 2, and since 2 is a rational number, the problem is solved.
the quadrilateral 
is cyclic. And it's circumcenter have the diameter 
.
,
is the midpoint of 
.
,where 
i
the one of the 
.
be the midpoint of 
Observe that 
so 
and 
so 
is cyclic. Let 
By sine rule on 
where 
is the foot of perpendicular on 
(since the radius of the nine point circle is half the circumradius). By sine rule on 
the circumradius of 
is![\[\dfrac{R \sin 2C}{2\angle MEF} = \dfrac{R}{2} \dfrac{\sin 2C}{\sin 2 \angle EBC} = \dfrac{R}{2} \dfrac{\sin 2C}{2 \sin \angle EBC \cos \angle BCE} = \dfrac{R}{2},\]](http://latex.artofproblemsolving.com/6/c/4/6c48f157e9076b6c7950fdf4c4b1488227d644da.png)
which is clearly a rational number.
