12. (15th All-Russian Mathematical Olympiad, 1989) Insert “+” or “-” signs between the numbers $1, 2, 3, \cdots, 1989$. What is the smallest non-negative number that can be obtained from the resulting sum?
12. Except for 995, all numbers $1,2,3, \cdots, 1989$ can be divided into 994 pairs: $(1,1989),(2,1988), \cdots,(994,996)$. Since the parity of the two numbers in each pair is the same, the result of the operation is always even regardless of how the “+” or “-” signs are placed before each pair. Since 995 is odd, the sum of the numbers $1,2,3, \cdots, 1989$ with any placement of “+” or “-” signs is always odd. Therefore, the smallest non-negative number that can be obtained is not less than 1.
On the other hand, the number 1 can be obtained in the following way:
$$
1=1+(2-3-4+5)+(6-7-8+9)+\cdots+(1986-1987-1988+1989) \text {. }
$$

Therefore, the smallest non-negative number that can be obtained from the sum is 1.