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.
This easy via the binomial theorem for the quantity is just 
,
and 
.
Y by HamstPan38825, Adventure10
Let 
, 
be two circles with radii 
and 
, and suppose the circles meet at 
and 
. Draw a line 
through intersecting , at 
and 
. Now allow the distance between the centers as well as the choice of to vary. Find the length of 
when the area of 
is maximized.
, 
be two circles with radii 
and 
, and suppose the circles meet at 
and 
. Draw a line 
through intersecting , at 
and 
. Now allow the distance between the centers as well as the choice of to vary. Find the length of 
when the area of 
is maximized.
.
,
since both are isosceles. Spiral similarity gives 
.
can be written as ![\begin{align*}[AO_1O_2]\left(\dfrac{AB}{AO_1}\right)^2=\left(\dfrac12 R_1 R_2\sin\angle O_1AO_2\right)\left(\dfrac{AB}{R_1}\right)^2=2R_1R_2\sin\angle O_1AO_2\sin^2\theta,\end{align*}](http://latex.artofproblemsolving.com/5/f/9/5f979ba302ed9f8ce4a3fd643c3db53168b2aeaf.png)
where the last equality is a result of the Extended Law of Sines. Maximizing ![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
is thus equivalent to maximizing both 
and 
.
and the latter is maximized by letting 
.
and 
,
is a kite, and so 
.
.![\[AX=\dfrac{AO_1\cdot AO_2}{O_1O_2}=\dfrac{R_1R_2}{\sqrt{R_1^2+R_2^2}}\implies AD=\boxed{\dfrac{2R_1R_2}{\sqrt{R_1^2+R_2^2}}}.\]](http://latex.artofproblemsolving.com/b/0/f/b0fc93a45f4282f5f97424ee46b52fdb24c2c82d.png)
,
and 
.
,
,
being the diameter of 
and 
;
,
.
,![\[\frac{2R_1R_2}{\sqrt{R_1^2 + R_2^2}}\]](http://latex.artofproblemsolving.com/7/9/1/79127d6795b5c18106b179f99385bb6a02234ae7.png)
![$[ABC]=\tfrac12 \sin{\angle CAB} \cdot |AB| \cdot |CA| \leq \tfrac12 \cdot 1 \cdot (2R_1) \cdot (2R_2)$](http://latex.artofproblemsolving.com/6/f/0/6f04b1191bee9930f6c1395ac223e24e84160016.png)
.
,
and 
.
.
.![\[ 2R_1R_2 = [ABC]=\frac12 |BC| \cdot |AD| \iff |AD| = \frac{2R_1R_2}{\sqrt{R_1^2+R_2^2}}. \]](http://latex.artofproblemsolving.com/f/a/b/fab563c108fbca19996ee93edbf19ab738085890.png)
And that's it.
