1974
Problem - 3831
Numbers $1,2,\cdots, 1974$ are written on a board. You are allowed to replace any two of these numbers by one number which is either the sum or the difference of these numbers. Show that after $1973$ times performing this operation, the only number left on the board cannot be $0$.
There are $987$ odd numbers on the board in the beginning. Every time the operation is performed, the number of the odd numbers left will be still odd. Hence, the leftover cannot be $0$ because if so means the number of odd number is even.