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 
.
be prime numbers such that their sum isn't divisible by 
.
positive integer quadruples satisfy:
Y by Adventure10, Mango247
For three points 
let 
be the circumcircle radius of the triangle 
If 
is a triangle with incircle centre 
then we have:
![\[ \frac{1}{R_{ABI}} + \frac{1}{R_{BCI}} + \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} + \frac{1}{\bar{BI}} + \frac{1}{\bar{CI}}.\]](//latex.artofproblemsolving.com/0/8/4/0841e84d09160d41dd2cb232ad23d88747428d48.png)
let 
be the circumcircle radius of the triangle 
If 
is a triangle with incircle centre 
then we have:![\[ \frac{1}{R_{ABI}} + \frac{1}{R_{BCI}} + \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} + \frac{1}{\bar{BI}} + \frac{1}{\bar{CI}}.\]](http://latex.artofproblemsolving.com/0/8/4/0841e84d09160d41dd2cb232ad23d88747428d48.png)
and 
,
.
and 
,
inside triangle 
(not necessarily 
)
