[/quote]
,
such that 
is a divisor of 
.
,
are equal, and they intersect at 
.
intersect at point 
lying inside triangle 
,
intersect at point 
lying inside triangle 
.
.
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 
.
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 
satisfying 
,![\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]](http://latex.artofproblemsolving.com/f/e/8/fe85cf52b883c214806ab837a2eb25cebb511d7b.png)
satisfy that: 
and natural number 
Y by
A circle is inscribed in an acute angle with vertex at point 
, tangent to the sides of the angle at points 
, 
. Prove that the length of an arbitrary segment, completely located inside the region, bounded by segments 
, 
and the smaller circular arc 
, is no more than .
, tangent to the sides of the angle at points 
, 
. Prove that the length of an arbitrary segment, completely located inside the region, bounded by segments 
, 
and the smaller circular arc 
, is no more than .
