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.
be positive real numbers such that 
Prove that we have![\[(a+b)(b+c)(c+a)\geq 8.\]](http://latex.artofproblemsolving.com/2/a/6/2a63f19de634fce70ff4ebc02ad13d442a20c378.png)
be the lengths of the sides of the triangle. Prove that:
such that 
Prove that 
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be a function such that
for all 
. Find all such functions 
.
be a function such that
for all 
. Determine all such functions .
. Find all 
such that for all 
be the set of real numbers. Find all functions 
where there exist a real constant 
such that 
holds for all reals 
, 
, 
that satisfy 
.
with its center 
. There are two fixed points 
and one moving point 
on . The midpoint of the line segment 
is 
. 
is a fixed point on . Line 
intersects
at 
, and line 
intersects 
at 
. such that is always tangent to 
when varies.
such that
be a scalene triangle with orthocenter 
and circumcenter . Denote by , 
the midpoints of 
, 
. Suppose the circle 
with diameter meets the circumcircle of at 
, and meets line 
at a point 
. The tangent to at 
meets line 
at 
. Show that the circumcircles of 
and 
intersect at a point 
on 
.
such that
for all 
) for all their time and support. We received immense help from TestX01 for the proof of claim 2.19 and it's associated lemma.
whose diagonals intersect at 
The bisector of the angle 
and that of 
intersect each other at 
Moreover, 
such that: 
.
