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 
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)
Y by
Let 
be a convex quadrilateral inscribed in a circle with center at point 
and radius 
(through 
here and in what follows we denote a circle with center at the point and radius ). Consider four circles 
, 
, 
, 
, and denote thus their points of intersection are different from the point : 
, 
, 
, 
. Prove that 
is a parallelogram.
be a convex quadrilateral inscribed in a circle with center at point 
and radius 
(through 
here and in what follows we denote a circle with center at the point and radius ). Consider four circles 
, 
, 
, 
, and denote thus their points of intersection are different from the point : 
, 
, 
, 
. Prove that 
is a parallelogram.This post has been edited 1 time. Last edited by parmenides51, Jun 29, 2021, 10:14 PM
