AoPS Academy
. Let 
be a subset of 
such that no sum of three distinct elements of is divisible by 
. What is the maximum value of 
.
that satisfy the equation 
.
be a circle, and let a line 
intersect at two points, 
and 
Circles 
and 
are internally tangent to at points 
and 
respectively, and both are tangent to at a common point 
Similarly, circles 
and 
are externally tangent to at and respectively, and are tangent to at points 
and 
respectively. is 
the segment 
has a length of 
and 
find the length of segment 
,
,
, 
, 
is not greater than 
.
and 
Prove that
.
Prove that
and 
Prove that
.
.
