such that 
for all reals 
.
Y by
Positive real numbers 
are written on the blackboard. A move consists of choosing two numbers 
and 
on the blackboard, erasing them and writing the number 
on the blackboard. After 
moves, only one number 
will remain on the blackboard. Prove that
![\[
c<2024 (a_1+a_2+\ldots+a_{2024}).\]](//latex.artofproblemsolving.com/0/2/5/025fc95f5cb5420adaef827c156811aa8291438e.png)
are written on the blackboard. A move consists of choosing two numbers 
and 
on the blackboard, erasing them and writing the number 
on the blackboard. After 
moves, only one number 
will remain on the blackboard. Prove that![\[
c<2024 (a_1+a_2+\ldots+a_{2024}).\]](http://latex.artofproblemsolving.com/0/2/5/025fc95f5cb5420adaef827c156811aa8291438e.png)
holds thus, 
doesn't increase. This implies 
by Cauchy-Schwarz and there is no equality case
it has a same value for each 
( it is constant, does not depend on the selection of the numbers). The main idea is this . Remain part is not hard . 
inequalities help)
