[/quote]
,
,such that 
P
be a positive integer. Show that there exist a polynomial 
with integer coefficient that satisfy the following
for each 
,
that satisfy the equation
be reals such that 
Prove that
Let 
Prove that
and 
.
be the point on the line 
.
and 
intersect at point 
such that 
.
is isosceles.
of positive integers with ![\[ a^p=b!+p. \]](http://latex.artofproblemsolving.com/9/f/6/9f674dabcb364872cef4153198e2e362ae40a2fd.png)
.
be a connected graph and let 
be two disjoint subsets of its vertices, such that there are no edges between them. Given that 
has 
connected components and 
has 
?
,
,
.
be the circle which touches 
,
lies outside of the excentral triangle of 
.
and 
cyclically. Let 
be the circle externally tangent to 
to be the least positive integer for which the following holds:
into 
disjoint subsets 
,
.
for which![\[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\]](http://latex.artofproblemsolving.com/2/0/9/2092eb944c746d1449e7865f9da4e0065b75bdef.png)
so that 
for all 
.
Y by
Circle 
touches the circumcircle of triangle 
at point 
, and it touches the sides 
and 
at points 
and 
, respectively. Rays 
and 
intersect at point 
. The tangent line at point to the circle meets the segment 
at point 
. The line segment 
intersects the circumcircle of triangle 
at point 
. Prove that the circumscribed circles of triangles and 
are tangent.
touches the circumcircle of triangle 
at point 
, and it touches the sides 
and 
at points 
and 
, respectively. Rays 
and 
intersect at point 
. The tangent line at point to the circle meets the segment 
at point 
. The line segment 
intersects the circumcircle of triangle 
at point 
. Prove that the circumscribed circles of triangles and 
are tangent.
is the intersection of 
and 
.
.
is external bisector of 
,
