2021 OIM Problems/Problem 5
Problem
For a finite set 
of integers, we define 
to be the sum of the elements of . Find two nonempty sets 
and 
, whose intersection is empty and whose union is the set 
, such that the product 
is a perfect square.
Solution
The solution to the equation is 
and 
; we can simply consider removing numbers to find the sets themselves, which just so happen to be
![\[A=\{x|x\in\mathbb{Z},x\in[1,1074]\cup[1076,1762]\}\]](http://latex.artofproblemsolving.com/d/8/1/d813c828903403d28ffb9c4d417c381d0f2da4f4.png)
![\[B=\{x|x\in\mathbb{Z},x\in\{1075\}\cup[1763,2021]\}\]](http://latex.artofproblemsolving.com/a/2/c/a2c219fd9b641fb4f992710dde0bd7146befc2f3.png)
