[/quote]
,
.![\[
(2^x - 1)(5^x - 1) = y^n
\]](http://latex.artofproblemsolving.com/f/8/f/f8f013a96863bc5f39aa23edb98eed9aec54a17f.png)
have no positive integer solution 
.
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x^2 + y) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/c/7/c/c7cd7c9a4d4372afc23e4503bb1c05fc7d07e788.png)
such that:
be a positive integer, and let 
denote the integers modulo 
satisfying 
for all 
.
such that 
divides 
for any integers 
and there exists a polynomial 
for all 
.
is call 
if it satisfies the following properties:
and 
for all indices 
.
.
for which: Every 
.
such that 
then find minimal value of 
has circumcircle 
and incircle 
,
at 
,
meets 
at 
,
meets 
meets 
.
is a point on 
are colinear. Let 
be the intersection of 
and 
is tangent to 
Y by Adventure10, Mango247
Let 
be two circles in the plane intersecting at two distinct points. Let 
be the midpoint of a variable chord 
of 
with the property that the circle on as diameter meets 
at a point 
such that 
is tangent to . Find the locus of .
be two circles in the plane intersecting at two distinct points. Let 
be the midpoint of a variable chord 
of 
with the property that the circle on as diameter meets 
at a point 
such that 
is tangent to . Find the locus of .
and 
,
and 
are tangent to 
,
be the centers of 
and 
,
the midpoint of 
and 
.
or 
.
