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 
be points on a plane such that 
, where 
is a positive integer. Let be the set of all points such that 
, where 
is a real number. The path that traces is continuous, and the value of is minimized. Prove that is rational for all positive integers 
are taken on sides 
, 
, 
, 
of rectangle 
, respectively. Draw parallels to the sides of passing through 
and create the rectangle 
as shown in the figure. Find the area of 
given that 
cm
and 
cm.
and 
Prove that
.
Prove that
and 
Prove that
.
.
![$[A_1B_1C_1D_1]$](http://latex.artofproblemsolving.com/f/c/d/fcd615cbae4c5c7635f440418d2fe6cd966ed299.png)
gives 
,
.
.