AoPS Academy
and points 
and 
on sides 
and 
, respectively, such that 
. Let 
and 
be intersection points of lines 
and 
with circumcircle 
of triangle 
, and 
and 
intersection points of lines 
and 
with side 
. If 
, prove that circle contains intersection point of angle bisector of 
with
such that 
divides 
for all positive integers 
and 
with 
.
such that 
Follows for all reals 
.
be a positive integer, and let 
denote the sum of the positive divisors of 
. Prove that the 
smallest positive integer relatively prime to is at least , and determine for which equality holds.
and 
are positive integers such that 
is perfect 33rd power of a positive integer, prove that 
is also a perfect 33rd power!
this time. Multiplying the congruence by 
,
,