Difference between revisions of "2021 OIM Problems/Problem 5"
(Created page with "== Problem == For a finite set <math>C</math> of integers, we define <math>S(C)</math> to be the sum of the elements of <math>C</math>. Find two nonempty sets <math>A</math>...") |
(solution) |
||
| Line 4: | Line 4: | ||
== Solution == | == Solution == | ||
| − | {{ | + | The solution to the equation is <math>S(A)=491103</math> and <math>S(B)=1552128</math>; we can simply consider removing numbers to find the sets themselves, which just so happen to be |
| + | <cmath>A=\{x|x\in\mathbb{Z},x\in[1,1074]\cup[1076,1762]\}</cmath> | ||
| + | <cmath>B=\{x|x\in\mathbb{Z},x\in\{1075\}\cup[1763,2021]\}</cmath> | ||
| + | |||
| + | ~ [https://artofproblemsolving.com/wiki/index.php/User:Eevee9406 eevee9406] | ||
== See also == | == See also == | ||
https://olcoma.ac.cr/internacional/oim-2021/examenes | https://olcoma.ac.cr/internacional/oim-2021/examenes | ||
Latest revision as of 17:31, 15 April 2025
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)
