Write the numbers on the blackboard. It is allowed to erase any two numbers and write down their sum or difference, repeating this procedure until only one number remains on the blackboard. Prove that this number cannot be zero.
Problem 995
Official solution
[Proof] Consider the number of odd numbers on the blackboard.
After one operation, if it is an odd number and an even number, then by the sum and difference of an odd number and an even number being odd, one odd number and one even number are erased to get one odd number. In this case, the number of odd numbers remains unchanged.
If it is two odd numbers, then by the sum and difference of two odd numbers being even, two odd numbers are erased to get one even number, and similarly, two even numbers are erased to get one even number, in which case, the number of odd numbers decreases by 2 or remains unchanged.
From the above, after the operation, the parity of the number of odd numbers on the blackboard does not change.
Since there are initially odd numbers on the blackboard, i.e., an odd number of odd numbers, after several operations, there must be an odd number of odd numbers left on the blackboard, and thus it cannot be 0.