,
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 
.
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
Y by
Let 
be a scalene acute triangle, where 
is the internal bisector of 
and 
is the midpoint of 
. Let the internal bisectors of 
and 
intersect 
and 
in 
and 
, respectively. Prove that the circumcircle of 
is tangent to if and only if 
belongs to it.
be a scalene acute triangle, where 
is the internal bisector of 
and 
is the midpoint of 
. Let the internal bisectors of 
and 
intersect 
and 
in 
and 
, respectively. Prove that the circumcircle of 
is tangent to if and only if 
belongs to it.
so 
so 
then 
passes through midpoint of arc 
for the second time. Let 
be the midpoint of arc 
meet 
.
would be cyclic so 
