be non-negative real numbers such that 
Prove that
and 
Find the minimum 
be a positive integer and 
be real numbers such that ![\[a_1+a_2+\dots+a_n=0.\]](http://latex.artofproblemsolving.com/0/f/d/0fde8d75cebc35103ae9acf8ebe393653d336d60.png)
Define the set 
by![\[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\]](http://latex.artofproblemsolving.com/6/9/7/6974ebf20c44bcb4d52d85adc0805e28aafa5d6e.png)
Prove that, if ![\[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]](http://latex.artofproblemsolving.com/b/e/3/be3e65a633dc2ac092a264968f1cf8514d3d6124.png)
Y by
In 
, the incircle 
is tangent to the sides 
and 
at 
and 
respectively. Let 
be the unique circle through 
and 
which is tangent to , the line through 
and the tangency point of and meets again at 
. Points 
and 
are defined analogously. Prove that
a)
concur at a single point, which we call 
, and
b)
are collinear and the distances between them satisfy 
, where 
and 
, respectively, denote the circumradius and inradius of 
.
, the incircle 
is tangent to the sides 
and 
at 
and 
respectively. Let 
be the unique circle through 
and 
which is tangent to , the line through 
and the tangency point of and meets again at 
. Points 
and 
are defined analogously. Prove thata)
concur at a single point, which we call 
, andb)
are collinear and the distances between them satisfy 
, where 
and 
, respectively, denote the circumradius and inradius of 
.
