be reals such that 
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.
This easy via the binomial theorem for the quantity is just 
,
and 
.
Y by Adventure10, Mango247
Let 
be the intersection of the diagonals of the cyclic quadrilateral 
. Let 
and 
denote the respective midpoints of the sides 
and 
. Prove that the line through which is perpendicular to 
, the line through which is perpendicular to 
and the line through which is perpendicular to 
are concurrent.
be the intersection of the diagonals of the cyclic quadrilateral 
. Let 
and 
denote the respective midpoints of the sides 
and 
. Prove that the line through which is perpendicular to 
, the line through which is perpendicular to 
and the line through which is perpendicular to 
are concurrent.
is sufficient (so that's a more general condition). Am I right? (I'll post my solution later.)
be the foot of the perpendicular from 
be the foot of the perpendicular from 
and 
intersect at 
.
.
and 
are similar. Hence we have (using similarity of triangles) that 
.
.
is cyclic, we get 
.
is the foot of the perpendicular on 
from 
is the foot of the perpendicular on 
being the foot of the perpendicular from 
and 
cross at 
.
is cyclic and angle chasing to get 
showing 
collinear.
was collinear to 
with 
.
are collinear. Notice that it would be enough to prove that 
is a cyclic quadrilateral since then 
as the both subtend the same arc and therefore 
which implies that 
and therefore 
is cyclic. Thus, by the Power of a Point theorem we have that
and 
are similar. Moreover, since 
and 
are similar. From this we see that 
but we also know that 
since they subtend the same arc. Therefore, 
which shows that 
is cyclic. Using the Power of a Point theorem again we have that
which we combine with the first equation to get
which shows that 
.
