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 an acute triangle 
, point 
is on the segment 
such that 
and 
. The line that is parallel to the bisector of 
and passes the point meets the segment 
at point 
. Prove, if 
, 
.
, point 
is on the segment 
such that 
and 
. The line that is parallel to the bisector of 
and passes the point meets the segment 
at point 
. Prove, if 
, 
.
,
.
is a 36-72-72 triangle. Thus, 
,
,
.
