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 
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 
.
+1 w
and a constant 
, it is given that 
and 
are also friendly polynomials. Prove that 
.
lying outside the circle 
, two tangents are drawn touching at points 
and 
. A point 
is chosen on the segment 
. Let points 
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects again at point 
(
). Prove that the line 
passes through the centroid of triangle 
.
be an acute-angled triangle with 
and let 
be the foot of the
-angle bisector on 
. The reflections of lines 
and 
in line meet and at points
and 
respectively. A line through meets and at 
and 
respectively such that and 
while lies strictly between 
and . Prove that the circumcircles of
and 
are tangent to each other.
. If 
and 
, show that the area of 
is not less than the area of 
. If 
and 
, show that the area of is not less than the area of 
. If 
and 
, show that the area of 
is not less than the area of 
.
, at least one of the three must hold: 
, 
, or 
. Otherwise 
which is contradiction.
then ![$[ABC]=\frac12 AB\times AC\times \sin\angle A\geq\frac12 DE\times DF\times\sin\angle D=[DEF]$](http://latex.artofproblemsolving.com/5/d/2/5d29a03577f3742076794563bc717eb89af3958e.png)
the area of triangle with sides 
. Then we want 
for 
, 
, 
.
, 
.
![\[(a,b,c)\ge(a,b,f)\ge(a,e,f)\ge(d,e,f)\]](http://latex.artofproblemsolving.com/2/f/5/2f5fba6ad1a9a1905f2b04d976837a18759e5193.png)
and we're done.
and 
.
.
Where 
such that :
