1976 IMO Problems/Problem 5
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
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: 
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