be real numbers . Prove that
![$$-\frac{1}{18}\leq \frac{ab+a+b+3}{(a^2+3)(b^2+3)}\leq \frac{2+\sqrt[3] 2+\sqrt[3] {4}}{12} $$](http://latex.artofproblemsolving.com/9/9/9/9997397eb38cd4bceec2ff8c85951c78e2797d0f.png)
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
and 
.
Y by Adventure10, Mango247
On the plane are given four distinct points 
on a line 
in this order, at the mutual distances 
.
(a) Construct (if possible) a point
outside line such that 
.
(b) Prove that such a point exists if and only if 
on a line 
in this order, at the mutual distances 
.(a) Construct (if possible) a point
outside line such that 
.(b) Prove that such a point
exists if and only if 
