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 the orthocenter of an acute angled triangle 
. Circumcircle of the triangle and the circle of diameter ![$[AH]$](http://latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
intersect at point 
, different from 
. Let 
be the midpoint of the small arc 
of the circumcircle of the triangle and let 
the midpoint of the large arc of the circumcircle of the triangle 
Prove that points 
are concyclic.
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
such that for any integer 
there exists an integer 
for which 
is divisible by .
are distinct prime numbers which satisfy
for natural number . Find all values of .
, point is the midpoint of , 
the point of intersection of the tangents at points 
and 
of the circumscribed circle of , is the midpoint of the segment 
. The segment 
meets the circumcircle at the point 
. Prove that 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
such that 
that satisfy the equation
with incenter 
. The incircle of triangle is tangent to at 
. Let and be points on the side BC such that 
and 
, respectively. Let 
and 
be the incenter of triangles 
and 
, respectively. Prove that 
is the Euler line of triangle 
.
be a triangle and let be its incenter. Prove that at least one of 
is greater than or equal to the diameter of the incircle of .
be a triangle and let be its incenter. Prove that at least one of is greater than or equal to the diameter of the incircle of .
is the smallest angle
diameter
:
be reals such that 
and 
Prove that
Where 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
