denote the set of positive rational numbers. Find, with proof, all functions 
such that, for all positive rational numbers 
and 
we have ![\[f(x)=f(x+y)+f(x+x^2f(y)).\]](http://latex.artofproblemsolving.com/9/e/a/9ea71d2ee277dc3c689ba2d791b451e29353edb3.png)
.
,
be the intersections of the perpendicular bisectors of 
and 
,
and 
.
and 
and the point 
is given. Let 
be set of all 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
,
and 
are integers, 
.
is said to be valuable if the intersection of any two segments from 
is either empty, or is a segment of nonzero length belonging to 
are there such that 
and 
?
and 
.
be real numbers such that 
Prove that
Let 
Prove that
and 
, then we can deduce 
\le 2 $ ?
satisfying 
.![\[
0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.
\]](http://latex.artofproblemsolving.com/4/2/d/42d0b88388462f16dd99d928b94024e689ba7c2e.png)
be a sequence defined by 
.
if 
.
Y by
Point 
is the midpoint of the side 
of an acute triangle 
. Point 
lies on the segment , and points 
and 
are the centers of the circumcircles of 
and 
, respectively. Show that the midpoint of segment 
lies on the perpendicular bisector of segment 
.
is the midpoint of the side 
of an acute triangle 
. Point 
lies on the segment , and points 
and 
are the centers of the circumcircles of 
and 
, respectively. Show that the midpoint of segment 
lies on the perpendicular bisector of segment 
.
be the midpoint of segment 
tells us that 
,
,
then we get 
,
.
to 
,
are midpoints of the lines that join the corresponding vertices of these triangles, so we actually have that 
and the result follows.
