with 
.
as 
,
and 
as 
.
.
are there such that 
and 
?
, where 
is a real number. Prove that for all positive 
then prove
Y by Adventure10, Mango247
Nine points, no three of which lie on the same straight line, are located inside an equilateral triangle of side 
. Prove that some three of these points are vertices of a triangle whose area is not greater than 
.
. Prove that some three of these points are vertices of a triangle whose area is not greater than 
.![[asy]
size(5cm);
path p = (2,0)--(0,0)--(2,2sqrt(3))--(4,0)--(2,0)--(1,sqrt(3))--(3,sqrt(3))--(2,0);
draw(p);
[/asy]](http://latex.artofproblemsolving.com/8/0/6/80638db8842c4e9cd1e15984f6dda4b27e0d01e0.png)
.
points in (or on the border of) one of the small triangles.The triangle formed by those 
points must have area less than or equal to 
