[/quote]
,
,
satisfying
be a positive integer, and let 
denote all positive divisors of 
satisfying the equation 
.
and 
Prove that
.
and 
but 
because 
.
and 
are perfect squares. 
.
it can draw the line which forms the interior angle bisector of 
.
in the plane . Prove that Euclid can use the splitter several times to draw the centre of circle passing through 
and 
.
,
,
?
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 
.
be the center of the circle tangent to sides 
,
,
and 
.
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 
.
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)
Y by
A circle with center 
of radius 
, touches the sides 
and 
of an acute triangle 
at points 
and 
, respectively. The straight line passing through the point parallel to , intersects the ray 
at the point 
. On the ray 
, the point 
is chosen so that 
. Find the length of segment if 
.
of radius 
, touches the sides 
and 
of an acute triangle 
at points 
and 
, respectively. The straight line passing through the point parallel to , intersects the ray 
at the point 
. On the ray 
, the point 
is chosen so that 
. Find the length of segment if 
.
