[/quote]
,
be the orthocenter of an acute angled triangle ![$[AH]$](http://latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
intersect at point 
,
.
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 
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 Adventure10, Mango247
A triangle 
is inscribed in a circle 
. Points 
and 
are the midpoints of the arcs 
and 
respectively, and 
is an arbitrary point on the arc 
(not containing 
). Points 
and 
are the incenters of the triangles 
and 
, respectively. If the circumcircle of triangle 
meets again at 
, prove that triangles 
and 
are similar.
is inscribed in a circle 
. Points 
and 
are the midpoints of the arcs 
and 
respectively, and 
is an arbitrary point on the arc 
(not containing 
). Points 
and 
are the incenters of the triangles 
and 
, respectively. If the circumcircle of triangle 
meets again at 
, prove that triangles 
and 
are similar.
and 
.
