The positive integer xxx which, when added to 1000, gives a sum greater than the product obtained when it is multiplied by 1000, satisfies the inequality:
1000+x>1000×x1000+x>1000\times x1000+x>1000×x
Solving the inequality:
1000+x>1000x1000+x>1000x1000+x>1000x
1000>1000x−x1000>1000x-x1000>1000x−x
1000>999x1000>999x1000>999x
1000999>x\frac{1000}{999}>x9991000>x
x<1.001x<1.001x<1.001
Since xxx is a positive integer, the only possible value is 111. Therefore, the positive integer is 111.
