33. Someone arrives in the city with very interesting news and after 10 minutes reports it to two people. Each of the newly informed people, after 10 minutes, reports it to two more (who have not yet heard it) and so on. *). How long will it take for the entire city to know this news, if the city has three million residents?
33. It is easy to see that the number of people who were told the news at the end of the 1st ten-minute period is 2, the number of people who were told the news after the 2nd ten-minute period is 4, and generally, the number of people who were told the news at the end of the $k$-th ten-minute period is $2^{k}$.

The number of all people who will know the news after $10 k$ minutes is equal to the number of people who learned this news after 0 minutes (i.e., 1 person), plus the number of people who learned the news after 10 minutes (2 people), plus the number of people who learned the news after 20 minutes (4 people), and so on up to the people who learned the news after $k \cdot 10$ minutes ($2^{k}$ people),

Therefore, this number of people is

$$
S_{k}=1+2+4+\ldots+2^{k}
$$

We see that $S_{k}$ is the sum of the first $k+1$ terms of a geometric progression with the first term 1 and common ratio 2. Therefore,

$$
S_{k}=\frac{1 \cdot 2^{k+1}-1}{2-1}=2^{k+1}-1 .
$$

Now, to solve the problem, we need to find the smallest $k$ for which $S_{k}$, i.e., the number of all people who know the news after $k \cdot 10$ minutes, will be greater than 3000000 (the total population of the city). It is easy to see that the smallest $k=21$, since

$$
2^{22}-1=4194303
$$

Thus, all residents of the city will know the news after 210 minutes $=$ $=3$ hours 30 minutes.