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 ![\[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\]](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 HWenslawski, oralayhan, Amir Hossein
In a triangle 
with 
, we define 
as the excenter with respect to 
. The lines 
and 
intersect in a point 
. Let 
be the circumcenter of 
. Prove that
![\[\frac{1}{|KA|} = \frac{1}{|KD|} + \frac{1}{|KE|}.\]](//latex.artofproblemsolving.com/b/b/7/bb70ee0d15a3558b81d46af6c401e48b3ea85fbc.png)
with 
, we define 
as the excenter with respect to 
. The lines 
and 
intersect in a point 
. Let 
be the circumcenter of 
. Prove that![\[\frac{1}{|KA|} = \frac{1}{|KD|} + \frac{1}{|KE|}.\]](http://latex.artofproblemsolving.com/b/b/7/bb70ee0d15a3558b81d46af6c401e48b3ea85fbc.png)
This post has been edited 3 times. Last edited by Amir Hossein, Mar 12, 2024, 9:22 PM
.
.
Also,
Therefore, it suffices to show that
, which is correct. Done.
then 
( with signed distance)
the symmetric of 
.
