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 
.
and 
. Prove that 
. Prove that
Y by Adventure10, Mango247
A quadrilateral 
inscribed in a circle 
. Angle bisector of 
and 
meet at 
and angle bisector of 
and 
meet at 
. Let 
be the midpoint of the arc 
of which does not contain 
and 
be the midpoint of arc 
which does not contain 
. Prove that 
.
inscribed in a circle 
. Angle bisector of 
and 
meet at 
and angle bisector of 
and 
meet at 
. Let 
be the midpoint of the arc 
of which does not contain 
and 
be the midpoint of arc 
which does not contain 
. Prove that 
.
;
,
(my drawing got 
)
belong to bisector of 
,
.
.
is the excentre and incentre of 
respectively, hence 
angle bisector of 
,
into eachother in the same speed. Let they coincide at a point 
.
angle bisector at 
with same definition as before. This is very easy, because if 
,
is incentre, then 
and 
and so 
.
