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 
. Prove that
Equality holds when 
or 
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 a constant 
, it is given that 
and 
are also friendly polynomials. Prove that 
.
and are marked in the plane not lying on the line 
. For all right triangles 
with hypotenuse on , show that the circumcircle of triangle 
passes through a fixed point other than .
be the foot of perpendicular from to .It's well known that 
the other fixed point is the image of in inversion in with radius 
and 
.
and 
.
Where 
such that :
be on 
.
is fixed, so 
,
.
,
goes through a fixed point. Letting 
,
,
.
or 