[/quote]
,
.
, 
and 
.
.
such that 
is a divisor of 
.
,
are equal, and they intersect at 
and 
intersect at point 
lying inside triangle 
,
and 
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)
such that there exists a unique integer 
such that 
with the following property:![\[ n!\mid a^n + 1
\]](http://latex.artofproblemsolving.com/e/2/3/e23d39aab685888993d355d8c93d4e6475f53487.png)
(use in a problem for junior) to practice. Can anyone share to me?
side length and with center 
.
is drawn where point 
.
Y by
There are two circles on the plane 
and 
with centers 
and 
respectively touch externally at the point 
, with the radius of the circle larger than the radius of the circle 
. Consider a point 
such that the points , 
and 
are not collinear. 
and 
are tangents to the circle (
and 
are points of contact). Lines 
and 
intersect the second circle at points 
and 
, respectively. The point of intersection 
and tangent at the point 
to the circle is denoted by 
. Prove that the point 
lies on a fixed line when the point moves in a circle such that the points 
, and are not collinear.
and 
with centers 
and 
respectively touch externally at the point 
, with the radius of the circle larger than the radius of the circle 
. Consider a point 
such that the points , 
and 
are not collinear. 
and 
are tangents to the circle (
and 
are points of contact). Lines 
and 
intersect the second circle at points 
and 
, respectively. The point of intersection 
and tangent at the point 
to the circle is denoted by 
. Prove that the point 
lies on a fixed line when the point moves in a circle such that the points 
, and are not collinear.
at 
is the symmedian of 
.
it follows that 
is also symmedian of 
.
is harmonic, it follows that 
,
