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 a point on the perpendicular bisector of 
(see figure) and 
,
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 Adventure10, Mango247
Let 
be the set of all tetrahedra which satisfy:
(1) the base has area
,
(2) the total face area is
, and
(3) the angles between the base and the other three faces are all equal.
Find the element of which has the largest volume.
be the set of all tetrahedra which satisfy:(1) the base has area
,(2) the total face area is
, and(3) the angles between the base and the other three faces are all equal.
Find the element of
which has the largest volume.
