$\underline{\text { Stunjas L. }}$.

An ATM exchanges coins: doubloons for pistoles and vice versa. A pistole costs $s$ doubloons, and a doubloon costs $1 / s$ pistoles, where $s$ is not necessarily an integer. You can insert any number of coins of one type into the ATM, after which it will dispense coins of the other type, rounding the result to the nearest whole number (if there are two nearest numbers, the larger one is chosen).

a) Can it be that by exchanging some doubloons for pistoles, and then exchanging the received pistoles for doubloons, we end up with more doubloons than we started with?

b) If so, can it happen that the number of doubloons obtained will increase even more if we perform the same operation with them?
a) Let, for example, $s=3$. Exchanging 5 doubloons, we get 2 pistoles, and exchanging the pistoles, we get 6 doubloons.

b) Let $s1$ and after the first exchange, we get $n$ pistoles. As shown above, after two exchanges, we will not get more than $n$ pistoles, therefore, the number of doubloons after the fourth exchange will not be greater than after the second.

## Answer

a) Can; b) cannot.