Announce $A$.

$N$ friends simultaneously learned $N$ pieces of news, with each person learning one piece of news. They started calling each other and exchanging news.

Each call lasts 1 hour. Any number of news items can be shared in one call.

What is the minimum number of hours required for everyone to learn all the news? Consider three cases:
a) $N=64$,

b) $N=55$

c) $N=100$.
a) A piece of news known to one of the friends will be known to no more than two (including the first) after 1 hour, no more than four after the second hour, ..., and no more than 32 after the 5th hour. Therefore, it will take no less than 6 hours.

We will show that 6 hours are sufficient. The conversations can proceed according to the following scheme. We will number the participants with six-digit binary numbers. In the $k$-th hour, people whose numbers differ only in the $k$-th digit will converse (for example, in the 3rd hour, $a b c 0 d e$ will converse with $a b c 1 d e$). In this way, the number of pieces of news known to each person doubles every hour. (For example, after the 2nd hour, each person knows four pieces of news known to four participants whose numbers differ from his number in the first two digits.)

Remark. This method of conversation is only suitable for powers of two. Below is a method that works for any even numbers.

c) That 6 hours are insufficient is clear from a). We will show that 7 hours are sufficient. We will number the participants by elements of $\mathbf{Z}_{50} \times \{-1,1\}$. In the 1st hour, the participant with number $(x, y)$ will converse with $(x, -y)$, in the 2nd hour with $(x+1, -y)$, in the 3rd hour with $(x+3, -y)$, in the 4th hour with $(x+7, -y)$, in the 5th hour with $(x+15, -y)$, in the 6th hour with $(x+31, -y)$, and in the 7th hour with $(x+63, -y)$. In this way, the number of pieces of news each friend has doubles every hour (except the last). (We will number the pieces of news the same way as the friends who know them at the beginning. After the 1st hour, the participant with number $(0,0)$ knows all pieces of news with $x=0$, after the 2nd hour all pieces of news with $x=0,1$, after the 3rd hour all pieces of news with $x=0,1,2,3$; and so on.)

b) In the first hour, one of the participants does not converse with anyone. As seen from a), the others will need at least 6 more hours to learn his news.

We will divide the participants into two groups: 32 and 23 people. In the 1st hour, all members of the second group will converse with members of the first. Over the next 5 hours, the members of the first group will exchange news (according to the scheme from a) or c); as a result, each will know all the news). In the last hour, they will inform all the members of the second group with all the information.

## Answer

a) 6 hours, b) 7 hours, c) 7 hours.