Đáp án:

 `S={1000}.`

Giải thích các bước giải:

 `\frac{x-1}{999}-\frac{x+1}{1001}+\frac{x-2}{998}-\frac{x+2}{1002}+\frac{x-3}{997}-\frac{x+3}{1003}+\frac{x-4}{996}-\frac{x+4}{1004}=0`

`<=>(\frac{x-1}{999}-1)+(1-\frac{x+1}{1001})+(\frac{x-2}{998}-1)+(1-\frac{x+2}{1002})+(\frac{x-3}{997}-1)+(1-\frac{x+3}{1003})+(\frac{x-4}{996}-1)+(1-\frac{x+4}{1004})=0`

`<=>(\frac{x-1-999}{999})+(\frac{1001-x-1}{1001})+(\frac{x-2-998}{998})+(\frac{1002-x-2}{1002})+(\frac{x-3-997}{997})+(\frac{1004-x-4}{1004})=0`

`<=>\frac{x-1000}{999}+\frac{1000-x}{1001}+\frac{x-1000}{998}+\frac{1000-x}{1002}+\frac{x-1000}{997}+\frac{1000-x}{1004}=0`

`<=>\frac{x-1000}{999}-\frac{x-1000}{1001}+\frac{x-1000}{998}-\frac{x-1000}{1002}+\frac{x-1000}{997}-\frac{x-1000}{1004}=0`

`<=>(x-1000).(\frac{1}{999}-\frac{1}{1001}+\frac{1}{998}-\frac{1}{1002}+\frac{1}{997}-\frac{1}{1004})=0`

Ta nhận thấy:`\frac{1}{999}-\frac{1}{1001}+\frac{1}{998}-\frac{1}{1002}+\frac{1}{997}-\frac{1}{1004}\ne0`

Nên: `x-1000=0`

`<=>x=1000`

Vậy `S={1000}.`