AoPS Academy
is given where 
are positive reals. Prove that:
be a triangle and let 
be a circle in its plane passing through 
and 
Suppose there exist circles 
such that for 
is externally tangent to 
and passes through 
and 
where 
for all 
. Prove that 
and 
have common external tangents 
and 
meeting at 
. Suppose touches at 
and touches at 
. A circle 
through and intersects again at 
and again at 
, such that quadrilateral 
is convex.
and 
meet at point 
, while lines 
and 
meet at point 
. Show that , , are collinear.
points in a plane which are vertices of a convex polygon. Prove that the set of the lengths of the sides and the diagonals of the polygon has at least 
elements. because then easily follows for 
. is even. Now put in the set 
the greatest of all sides.
vertices, for 
.
such sorts of diagonals and each time we get one lenght that is greater than previous greatest. The statement follows.
)
vertices of an equilateral triangle of side 
, 
. Take the angle bisector at choose 
a point of it at distance from . If you do this for and , you obtain 
points in convex position that only define 
distances. Thus the problem is false.
, and probably 
. How about 
? There are three distinct distances.
-gons, and for even, also regular 
-gons with one vertex removed.
