AoPS Academy
be a positive integer that is not divisible by 
. Furthermore, let 
be a positive integer. Prove that none of the numbers of the form 
is a square number.
are such that the numer
is rational. Find all possible values of 
or show that there are infinitely many.
intersect at point 
. The points of tangency of the circumcircles of triangles 
and 
with their common external tangents lie on a circle 
. The points of tangency of the circumcircles of triangles 
and 
with their common external tangents lie on a circle 
. Prove that the centers of circles and coincide.
be a positive integer. Furthermore, let ![$x_1, x_2,\ldots, x_n \in [0, 2]$](http://latex.artofproblemsolving.com/b/2/a/b2a6731961601353e26224c285a32923fbb54eb3.png)
be real numbers subject to 
. Prove the inequality
When does equality hold?
be a real number such that the system 
has exactly one complex solution 
The sum of all possible values of can be written as 
where 
and 
are relatively prime positive integers. Find 
Here 
.So 
lies on 
and 
and 
.Let the angle bisector of cuts the line 
at 
.Now 
can be easily found and 
.So it's easy to show that 
. So 
.Now 
is the medial triangle of 
and the cevian 
is also the angle bisector of 
. Hence the result follows.
cut circumcircle 
of again at 
is feet of perpendicular from 
to 
Let 
be feet of perpendiculars from to 
is Simson line with pole . 
is kite 
. By Archimedes theorem, 
for 
and 
for 
is internal bisector of . Likewise, 
are internal bisectors of 
concurrent at incenter of 
.
are the midpoints of the sides 
.
,
.
are concurrent.
