1976 IMO Problems/Problem 5
Problem
We consider the following system with :
in which every coefficient is an element from the set Prove that there exists a solution for the system with the properties:
a.) all are integers
b.) there exists at least one j for which
c.) for any
Solution
First of all note that we have possible nonzero vectors such that are integers.
But can only assume different values, because if it is maximized/minimized by , we have that (if , it doesn't affect the sum, if it is , , and if it is , ).
From this we conclude that there are at most possible values for the vector .
But we have that:
We conclude that by the pigeonhole principle there are two distinct vectors being mapped to the same vector. Taking their difference we have a vector with the desired properties.
The above solution was posted and copyrighted by Jorge Miranda. The original thread for this problem can be found here: [1]
See also
1976 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |