such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
in triangle 
.
,
,
be the feet of the internal angle bisectors of 
,
,
respectively.
be the incenter of triangle 
,
be the foot of the perpendicular from 
.
is cyclic, then the center of its circumscribed circle lies on segment 
.
of one variable with integer coefficients such that if 
and 
are natural numbers such that 
is a perfect square, then 
is also a perfect square.
be a nonempty set of positive integers such that, for any (not necessarily distinct) integers 
and 
in 
is also in 
Y by Adventure10
Let 
be an isosceles triangle with 
.Take a point 
on the extention of 
from 
such that 
.The angle bisector of 
intersect with 
at 
and 
is midpoint of .The tangent from to circumcircle of triangle 
intersect with 
at 
.Prove that 
are concyclic.
be an isosceles triangle with 
.Take a point 
on the extention of 
from 
such that 
.The angle bisector of 
intersect with 
at 
and 
is midpoint of .The tangent from to circumcircle of triangle 
intersect with 
at 
.Prove that 
are concyclic.
be the midpoint of 
,
the intersection of 
the intersection of 
and 
,
,
and we get 
to 
,
,
,
