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 a triangle with circumcenter 
.
(see figure) and 
be the intersections of the perpendicular bisectors of 
,
and 
.
and 
and the point 
is given. Let 
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
The parallelogram 
and the rhombus 
have a common angle 
. It is known that 
. Let 
be the intersection point of of the lines 
and 
. Let 
be the intersection point of the lines 
and 
. Prove that the angle 
is right.
and the rhombus 
have a common angle 
. It is known that 
. Let 
be the intersection point of of the lines 
and 
. Let 
be the intersection point of the lines 
and 
. Prove that the angle 
is right.This post has been edited 2 times. Last edited by parmenides51, Jul 7, 2021, 6:13 PM
