AoPS Academy
. Prove that:
be reals such that 
Prove that 
Let 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.
be circumcircles of the cyclic pentagons 
The 
are oppositely similar, having equal angles. Let 
be circumcenter of the 
Because of the opposite similarity, 
is isogonal of 
with respect to the angle 
hence 
and similarly, 
Thus is the intersection of the C-altitude of the 
with the perpendicular bisector of 
passing through 
on the perpendicular bisector of 
(both 
), 
(both 
), 
is a parallelogram, so that the circumradius of the 
is equal to 
In exactly the same way, we can find that the circumradius of the 
with circumcenter 
is equal to 
If the bases 
and the circumradii 
of the 
are equal, the angles 
are also equal.
such that :
and 
lie on sides 
and 
of the convex quadrangle 
Let 
and 
be inscribed pentagons and 
Prove that 
then points 
are collinear and it is not difficult to verify that 
also holds true. 
then