[/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 
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 
of the circumscribed circle of 
.
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 
,
.
Y by Davrbek, Mathuzb, Adventure10
Triangle 
with 
given and point 
is chosen on the side 
. 
is a circle that touches 
and extensions of 
and 
with radius 
and center 
. 
is a circle that touches 
and extensions of 
and with radius 
and center 
. Let 
be intersection of the extension of and tangent at to circumference of 
. Prove that 
.
with 
given and point 
is chosen on the side 
. 
is a circle that touches 
and extensions of 
and 
with radius 
and center 
. 
is a circle that touches 
and extensions of 
and with radius 
and center 
. Let 
be intersection of the extension of and tangent at to circumference of 
. Prove that 
.
and 
.
.
.
be on the extension of 
such that 
.
is the external bisector of 
,
is cyclic, thus 
,
.
is centered at 
be the unique point such that 
and 
is convex.
It follows that 
From above we conclude that 
is tangent to 
and 
which completes the proof.
