Two numbers $a$ and $b(a>b)$ are written on the board. They are erased and replaced by the numbers ${ }^{a+b} / 2$ and ${ }^{a-b} / 2$. The same procedure is applied to the newly written numbers. Is it true that after several erasures, the difference between the numbers written on the board will become less than $\frac{1 / 2002}{}$?
Perform the operation twice and see what happens to the numbers.

## Solution

By performing the specified operation twice, we get the numbers $\frac{a}{2}, \frac{b}{2}$. This means that after $2 k$ operations, the numbers will be $2^{-k} a, 2^{-k} b$. But for sufficiently large $k$

$2^{-k} a-2^{-k} b<1 / 2002$

## Answer

Correct.