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and 
such that![\[
1^n + 2^n + 3^n + \cdots + n^n = x!
\]](http://latex.artofproblemsolving.com/6/4/b/64b5f5919fd3710cc15e96724918eae216c8e4f0.png)
be a cyclic convex quadrilateral and 
be its circumcircle. Let 
be the intersection of the diagonals of 
and 
. Let 
be the center of the circle tangent to sides 
, 
, and 
, and let 
be the midpoint of the arc of not containing 
and 
. Prove that the excenter of triangle 
opposite lies on the line 
.
that has 8097 distinct points with area of a triangle that has 3 points belong to all 
. Prove that there exists a triangle 
that has the area 
contains at least 2025 points that belong to ( each of that 2025 points can be inside the triangle or lie on the edge of triangle )X
and 
be the sidelengths of a triangle and 
be the area of that triangle. Prove that 
.
,
,
,![\[
\sum_{{\rm sym}}(m+n)(m+k)=\sum m^2 + 3\sum mn \ge 4\sqrt{3}\sqrt{mnk(m+n+k)},
\]](http://latex.artofproblemsolving.com/2/4/4/244cc3bc3a52fccc3e9e8287cc13f9a3dc637c49.png)
for 
.
,![\[
mn+nk+km\ge \sqrt{3mnk(m+n+k)}\iff (mn+nk+km)^2 \ge 3mnk(m+n+k),
\]](http://latex.artofproblemsolving.com/6/4/e/64edc7651236d64fb1decac63997799120beb139.png)
which is trivial, and follows by settting 
,
and 
,
.