[/quote]
,
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 
,
be the intersections of the perpendicular bisectors of 
and 
,
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 
rows of coins. There is 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on, up to 
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)
satisfy that: 
(use in a problem for junior) to practice. Can anyone share to me?
be a square with 
side length and with center 
.
is drawn where point 
lies on line 
.
and natural number 
Y by
It is given a circle with center 
and radius 
. 
and 
are two diameters. The lines 
and 
are tangent to the circle at the points 
and 
and intersect at point 
. 
and 
are the midpoints of the segments 
and 
. Prove that:
(a) the points
are concyclic.
(b) the heights of the triangle
intersect in the midpoint of the radius 
.
and radius 
. 
and 
are two diameters. The lines 
and 
are tangent to the circle at the points 
and 
and intersect at point 
. 
and 
are the midpoints of the segments 
and 
. Prove that:(a) the points
are concyclic.(b) the heights of the triangle
intersect in the midpoint of the radius 
.
