be an integer. The numbers 
are split into 
groups of 
.
quadruples 
such that they are all in different groups, 
and 
.
be positive real numbers so that 
.![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
be an acute triangle with orthocenter 
.
which is parallel to 
and tangent to the incircles of 
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 
