AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
, 
, 
and 
lie on a circle 
such that 
. The tangent line to at point intersects the tangent line to at and the line 
at 
and 
. The circle and the circumcircle of triangle 
intersect again at 
. Prove that 
.
and 
. Prove that
Equality holds when 
or 
+1 w and . Prove that
Equality holds when or 
intersect its circumcircle 
at 
. The tangets to from 
determine a triangle 
, where 
is relative to and so on. Prove that lines 
are concurrent at a point 
and that belongs to the euler line of .
be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points 
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at 
, and 
intersect 
at 
. Prove that lines 
and 
are either parallel or coincide.
. 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.
This easy via the binomial theorem for the quantity is just 
, but how do we arrive at this using the I-E-P?
satisfies 
. If 
, find the maximum possible value of 
.
and 
.
and 
.
Where 
such that :
denote the midpoints of 
,
be vectors on the unit sphere corresponding to 
,
Indeed, firstly observe that this center 
must be unique, say it corresponds to the vector 
.
which corresponds to the vector 
,
'
,
Notice that this point 
,
We know from Apollonius' Median Formula that 
With similar expressions for 
,
.