such that :
.
be reals such that 
Prove that 
Let 
Prove that 
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
colors, such that among any 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible.
Y by Adventure10, Mango247
Two circles of different radius with centers at 
& 
respectively touch each other externally at 
.
A common tangent not through touches the first circle at 
and second at 
.
The line through which is perpendicular to 
and the perpendicular bisector of 
meet at 
.
Prove that
.
& 
respectively touch each other externally at 
.A common tangent not through
touches the first circle at 
and second at 
.The line through
which is perpendicular to 
and the perpendicular bisector of 
meet at 
.Prove that
.
and 
at 
.
is the mid-point of 
be the mid-point of 
.
is the mid-point of 
(Considering 
)
is a trapezium 
)
is its mid-line.![\[\therefore AF=ML=\frac{1}{2}(BD+CE)=\frac{1}{2}(BA+CA)=\frac{1}{2}BC \Rightarrow BC=2AF\]](http://latex.artofproblemsolving.com/6/7/1/671077ba756692fcd05c42b317ae3b4f485c4ce2.png)
