[/quote]
,
,
satisfying 
,
?
and 
such that![\[
1^n + 2^n + 3^n + \cdots + n^n = x!
\]](http://latex.artofproblemsolving.com/6/4/b/64b5f5919fd3710cc15e96724918eae216c8e4f0.png)
be its circumcircle. Let 
be the intersection of the diagonals of 
and 
.
be the center of the circle tangent to sides 
,
,
opposite 
.
that has 8097 distinct points with area of a triangle that has 3 points belong to 
.
that has the area 
contains at least 2025 points that belong to 
crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least 
crosses.
be the incentre of acute triangle 
.
of 
,
and 
,
meets 
.
meets 
and 
meet again at 
and 
meet on the line through 
.
and 
Prove that
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
and 
Prove that
integers are such that 
is prime and 
.
such that 
.
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
.
be 
respectively. A circle with a center 
,
.
,
.
,
,
and lines 
are concurrent.
Y by Adventure10, Rounak_iitr
The quadrilateral 
is inscribed in the circle 
. Bisectors of angles 
and 
intersect at point 
, and bisectors of angles 
and 
intersect in point 
. Point 
is the center of this arc 
of the circle which does not contain points 
and 
. Point 
is the center of the arc 
of the circle , which does not contain points 
and 
. Prove that the points and lie on the line perpendicular to 
.
is inscribed in the circle 
. Bisectors of angles 
and 
intersect at point 
, and bisectors of angles 
and 
intersect in point 
. Point 
is the center of this arc 
of the circle which does not contain points 
and 
. Point 
is the center of the arc 
of the circle , which does not contain points 
and 
. Prove that the points and lie on the line perpendicular to 
.This post has been edited 4 times. Last edited by mruczek, Apr 21, 2018, 4:38 PM
are collinear.
and Q is the excenter of 
,
.
and tangent of circle at 
,
and 
,
and 
intersect at 
,
is parallel to 
,
