40. Given numbers: one and nine zeros. It is allowed to choose two numbers and replace each with their arithmetic mean. What is the smallest number that can end up in the place of one after a series of such operations?
84.40. Let's consider the smallest non-zero number $A$. Under the averaging operation, this value either does not decrease or is halved if the averaging is done with zero. From this, it follows that the value of $A$ will not decrease by more than $2^{9}=$ $=512$ times. Answer. $1 / 512$.