AoPS Academy
, let 
denote the number of its positive divisors (including 1 and itself). Determine all positive integers 
for which there exists a positive integer such that 
.
let 
be the foot of the altitude from 
to the side 
and 
, 
, 
be the incenter, -excenter, and 
-excenter, respectively. Denote by 
and 
the other intersection points of the circle 
with the lines 
and 
, respectively. Prove that 
.
such that![\[(a^2-b^2)(b^2-c^2) = abc.\]](http://latex.artofproblemsolving.com/7/1/8/7188261c40a45f5e15533100432d36abbf515efd.png)
. If this is still too hard assume in 
that 
is a prime.
is divided into rectangles with two cells. Two rectangles are called adjacent if they share a segment of length 1 or 2. In each rectangle the amount of adjacent with it rectangles is written.
be an incenter of triangle 
and let incircle touch sides 
and 
in points 
and 
, respectively. Lines 
and 
intersect line 
in points 
and 
, respectively. Prove that points , , 
and are concyclic
be an incenter of triangle and let incircle touch sides and in points and , respectively. Lines and intersect line in points and , respectively. Prove that points , , and are concyclic
so 
is cyclic. Hence 
and similarly 
,
is the Thales circle over 