3. A store sells 20 kilograms of cheese. The first 10 customers, after each purchase, the salesperson tells the other customers: “If the amount you buy is equal to the average of what the previous customers bought, is it possible?” If possible, after the next 10 customers buy, how much cheese will be left?
3. It is possible.

For $i \leqslant k<10$, if the first $k$ customers buy cheese $\frac{\dot{k}}{10+k}$, then after the $k$-th customer buys, the average amount of cheese sold to each customer is $\frac{1}{10+k}$, and $\frac{10}{10+k}$ remains. Therefore, it is still enough to sell to 10 customers.
Notice that the amount of cheese the $k$-th customer buys is
$$
\frac{k}{10+k}-\frac{k-1}{9+k}=\frac{10}{(10+k)(9+k)},
$$

These values are certain. When the 10th customer buys, the amount of cheese sold is $\frac{10}{10+10}=\frac{1}{2}$, which means that $10 \mathrm{~kg}$ of cheese remains.