be an integer and let 
be 
distinct subsets of 
.
such that the n subsets 
are also disjoint.
s
.
satisfies 
.
,
.
such that for all irrational numbers 
and 
,
where 
Y by Adventure10, Mango247
The semicircle with centre 
and the diameter 
is divided in two arcs 
and 
with ratio 
. 
is the midpoint of the radium 
. Let 
be the point of arc such that the area of the cuadrylateral 
is maximum. Find such area in fuction of the radium.
and the diameter 
is divided in two arcs 
and 
with ratio 
. 
is the midpoint of the radium 
. Let 
be the point of arc such that the area of the cuadrylateral 
is maximum. Find such area in fuction of the radium.![$[OBTM]=[OBM]+[MBT]$](http://latex.artofproblemsolving.com/a/5/b/a5b5af54ebd53191bc38b7bbb5cba1dc28b287e3.png)
![$[OBM]$](http://latex.artofproblemsolving.com/4/d/1/4d1ee1ff2dc7c2312970b09354c48d42fe31f7c3.png)
is always fixed it is not dependent on point 
of 
![$\displaystyle [MBT]=\frac{1}{2} (BM)(TH)$](http://latex.artofproblemsolving.com/4/8/8/488a6e0d4790b47ea8a099982d651a7fa67592d2.png)
,
.
,
.![$[OBTM]$](http://latex.artofproblemsolving.com/1/0/0/100e13acd31ccea91dbf37575ef200c5b053ffb2.png)
can be expressed as ![$[OBTM]=\frac{1}{2}[BM][OT]$](http://latex.artofproblemsolving.com/3/2/c/32cb0c88ab766dc56d1ffbedff40d52a283638f0.png)
,![$[OBTM]=\frac{1}{2}\left(r\frac{\sqrt{5+2\sqrt{2}}}{2}\right)\left(r\right)=\frac{\sqrt{5+2\sqrt{2}}}{4}r^2$](http://latex.artofproblemsolving.com/3/3/8/338630ee95ff63124a9ecfebd9304269003c0321.png)
