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.
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
such that 
with 
. Denote the orthocenter of 
as 
, the intersection of 
and 
as 
. Compute the length of 
.
be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer 
such that 
.
such that 
be a set of Natural numbers from 
to 
. (
) distinct elements from this set.
. If 
and 
, show that the area of 
is not less than the area of 
, and the pairwise comparison of the sums of elements of all its subsets (with the empty set defined as having a sum of 0), which amounts to 
inequalities, these given comparisons satisfy the following three constraints:
and 
, if the sums of elements of and are greater than those of subsets 
and 
respectively, then the sum of elements of the union 
is greater than that of 
.
that satisfies all these conditions?
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
, by 
the midpoint of , and by 
the orthocenter of . Let 
be the point of intersection of the circumcircle 
of the triangle and the half line 
, and 
be the point of intersection (other than 
) of the line 
and the circle . Prove that 
must hold.
the length of the line segment .)
such that 
(where 
), and a functional equation 
that satisfies 
for all , has a finite number of solutions. Find such .
is the function obtained by composing 
times, that is, 
)
are there such that 
and 
?
, where 
is a real number. Prove that for all positive , 
![[asy]
add(grid(4,3));
fill((0,3)--(4,3)--(2,2)--cycle);
fill((0,0)--(0,1)--(2,2)--(1,0)--cycle);
fill((2,2)--(3,0)--(4,0)--(4,1)--cycle);
[/asy]](http://latex.artofproblemsolving.com/f/b/2/fb29b865631cc6ddede429902646c4d1286f5e28.png)
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
.
