[/quote]
,
be the orthocenter of an acute angled triangle 
.![$[AH]$](http://latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
intersect at point 
,
.
be the midpoint of the small arc 
of the circumcircle of the triangle 
the midpoint of the large arc 
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 
the point of intersection of the tangents at points 
and 
.
meets the circumcircle 
.
:
be reals such that 
and 
Prove that
Where 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that 
that satisfy the equation
.
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
Y by
Let 
and 
be circles that intersect at two different points 
and 
. Let 
and 
be points on and such that 
and 
are tangent to and , respectively. Let 
be a point on such that 
and 
a point on such that 
. Let 
be the intersection point of 
with 
, and 
the intersection point of 
with 
. Prove that the circumcircle of the triangle 
is tangent to .
and 
be circles that intersect at two different points 
and 
. Let 
and 
be points on and such that 
and 
are tangent to and , respectively. Let 
be a point on such that 
and 
a point on such that 
. Let 
be the intersection point of 
with 
, and 
the intersection point of 
with 
. Prove that the circumcircle of the triangle 
is tangent to .
