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 ImSh95
Triangle 
is right at 
.
a) Show that the three ellipses that have foci at two vertices of the triangle and pass through the third have a point in common.
b) Show that the principal vertices of the ellipses in part (a), that is, the points at which each ellipse intersects the line joining its foci, form two collinear point themes.
is right at 
.a) Show that the three ellipses that have foci at two vertices of the triangle and pass through the third have a point in common.
b) Show that the principal vertices of the ellipses in part (a), that is, the points at which each ellipse intersects the line joining its foci, form two collinear point themes.
,
.
,
in the points 
and 
.
,
in the points 
and 
.
,
in the points 
and 
.
.
are collinear, the points 
are collinear.
