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 it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
such that 
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
that satisfy the equation
be an integer and denote 
. For a function 
we denote Im 
, Fix 
and 
. Find all possible values of 
Im 
+ Fix + 
.
:
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)
