[/quote]
,
.
, 
and 
.
.
such that 
is a divisor of 
.
,
and 
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 
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 
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 Adventure10
Let 
be an acutangle triangle and 
any point on the 
side. Let 
be the symmetrical of in
and 
is the symmetrical relative to 
. 
straight 
intersects straight at 
, while straight 
intersects the line in 
. Prove that the points 
and are on the same circumference.
be an acutangle triangle and 
any point on the 
side. Let 
be the symmetrical of in
and 
is the symmetrical relative to 
. 
straight 
intersects straight at 
, while straight 
intersects the line in 
. Prove that the points 
and are on the same circumference.This post has been edited 1 time. Last edited by Zelderis, Nov 23, 2019, 1:16 PM
Similarily 
gives that 
as desired.
