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
Let 
be a point in the interior of an acute triangle 
and 
be a convex hexagon whose vertices lie on the circumcircle 
of the triangle 
Let 
be the second point where the circle passing through and tangent to at 
intersects the line 
The points 
and 
are defined similarly. Prove that
![\[ \min\left\{\frac{PA_1}{AA_1}, \: \frac{QB_1}{BB_1}, \: \frac{RC_1}{CC_1}\right\} \leq 1.\]](//latex.artofproblemsolving.com/0/9/9/099bc897ed0a72be225376a31c554946cce8ab99.png)
be a point in the interior of an acute triangle 
and 
be a convex hexagon whose vertices lie on the circumcircle 
of the triangle 
Let 
be the second point where the circle passing through and tangent to at 
intersects the line 
The points 
and 
are defined similarly. Prove that![\[ \min\left\{\frac{PA_1}{AA_1}, \: \frac{QB_1}{BB_1}, \: \frac{RC_1}{CC_1}\right\} \leq 1.\]](http://latex.artofproblemsolving.com/0/9/9/099bc897ed0a72be225376a31c554946cce8ab99.png)
be the center of 
Since 
and 
as 
and on the same side of the bisector of the line segment 
as 
Then 
Let 
be the circle passing through 
Then 
Since 
lies inside 
Hence 
