such that ![\[f(f(x)) + xf(xy) = x + f(y)\]](http://latex.artofproblemsolving.com/a/6/f/a6f84dd53eec985b90bad3df6580eeaf29d73195.png)
for all positive real numbers 
and 
.
be real positive number satisfy that: 
.
,
Find all integers 
is an integer.
Y by Adventure10, Mango247
Points 
and 
are marked in the plane not lying on the line 
. For all right triangles 
with hypotenuse on , show that the circumcircle of triangle 
passes through a fixed point other than .
and 
are marked in the plane not lying on the line 
. For all right triangles 
with hypotenuse on , show that the circumcircle of triangle 
passes through a fixed point other than .
be on 
.
is fixed, so 
,
.
in a fixed circle 
,
goes through a fixed point. Letting 
,
,
is fixed.
the other fixed point is the image of 
.
or 
