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be non-negative real numbers such that 
Prove that
+1 w
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 is not empty, then![\[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]](http://latex.artofproblemsolving.com/b/e/3/be3e65a633dc2ac092a264968f1cf8514d3d6124.png)
are marked on the side 
of a triangle 
so that 
. Prove that 
.
and the point circle 
are coaxial, so 
and 
are tangent at , Now the inversion at with radius 
composed with a reflection about the bisector of 
takes to 
, two points on 
with 
. Thus 
is an isosceles trapezium, so 
, implying the result.
meet the circumcircle 
of 
at 
, and let the line 
meet at 
.
.
.
, so arcs 
and 
of must be equal as parallel chords cut off equal arcs.
.
and 
Find the minimum 
.
, 
,
,
