AoPS Academy
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to .
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
be prime numbers such that their sum isn't divisible by 
. Find the all 
positive integer quadruples satisfy:
be an -sided polygon with centroid 
, inscribed in a circle. The lines 
, 
,
, 
intersect again the circle at 
, 
, 
, respectively. Prove that
be an -sided polygon with centroid , inscribed in a circle. The lines , ,, intersect again the circle at , , , respectively. Prove that
