[/quote]
,
.
, 
and 
.
.
such that 
is a divisor of 
.
,
are equal, and they intersect at 
.
intersect at point 
lying inside triangle 
,
and 
intersect at point 
.
.
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 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 
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
Let 
be an acute triangle, with 
. Let 
be the midpoint of 
, 
the foot of the altitude from 
, and 
be point on side 
such that 
. Show that the circumcircles of the triangles 
and 
are tangent.
be an acute triangle, with 
. Let 
be the midpoint of 
, 
the foot of the altitude from 
, and 
be point on side 
such that 
. Show that the circumcircles of the triangles 
and 
are tangent.This post has been edited 1 time. Last edited by parmenides51, Jul 30, 2021, 9:44 AM
