[/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 
.
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 
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?
side length and with center 
.
is drawn where point 
lies on line 
.
and natural number 
Y by
Consider a convex quadrilateral 
having no right angle and whose four vertices belong to the same circle. Let us denote by 
and 
the feet of the perpendiculars lowered by 
respectively on 
and 
. Let us denote by 
and 
the feet of the perpendiculars lowered respectively from 
on 
and 
. The convex quadrilateral formed by four points 
:
(a) is it necessarily a trapezoid?
(b) does it necessarily have two equal sides?
having no right angle and whose four vertices belong to the same circle. Let us denote by 
and 
the feet of the perpendiculars lowered by 
respectively on 
and 
. Let us denote by 
and 
the feet of the perpendiculars lowered respectively from 
on 
and 
. The convex quadrilateral formed by four points 
:(a) is it necessarily a trapezoid?
(b) does it necessarily have two equal sides?
This post has been edited 2 times. Last edited by parmenides51, Dec 18, 2020, 7:28 PM
