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be an equilateral triangle, and 
is a point on its incircle. Let 
be the circle tangent to 
passing through and 
. Similarly, let 
be the circle tangent to 
passing through and 
, and 
be the circle tangent to 
passing through and 
., , has a common tangent line. be a triangle, 
its circumcenter and 
its circumcircle. Let 
and 
be points on and 
, respectively, such that is the midpoint of 
. Let 
, with 
. Finally, let be the point on line such that 
. Prove that the lines 
and the tangent to at 
are concurrent and that 
.
and 
is tangent to the circumcircle of triangle through point . Points 
and are the midpoints of segments and respectively, then line 
and line 
intersect at points 
and 
respectively. Line 
and line 
intersect the circumcircle of triangle at points 
and 
respectively. Prove that , , and concyclic.
to be a subset of size 
of 
. Prove that there exist 
pairwise disjoint, non-empty subsets 
such that 
and 
be positive real numbers, such that 
. Prove that![\[ (a^2 - a + 1)(b^2 - b + 1)(c^2 - c + 1) \ge 1. \]](http://latex.artofproblemsolving.com/e/9/d/e9d9385c60d7f3d0854fe16687e0361713579740.png)
, let 
be the maximum number of points inside or on the sides of a regular 
-agon of side 
such that the distance between any two points is greater than . Prove that 
for 
.
and a polynomial 
with real coefficients of degree 
(
) such that has no real roots. Prove that the polynomial![\[
Q(x) \;=\; P(x) \;+\; a\,P'(x) \;+\; a^2\,P''(x) \;+\; \dots \;+\; a^n\,P^{(n)}(x)
\]](http://latex.artofproblemsolving.com/6/5/e/65e5e4eefcadd5d0019b4929b3472cdf203c13aa.png)
has no real roots.
and 
Prove that
Where 
and 
.
Equality holds when 
or 
or 
and 
Prove that
Let 
Prove that
Let 
and 
Prove that
Let 
Prove that
be real numbers such that 
Prove that
Let 
Prove that
then 
and 
to get 
at 
where 
at 
Prove that
at 
