$:$ Folkiore

It is known that among 63 coins there are 7 counterfeit ones. All counterfeit coins weigh the same, all genuine coins also weigh the same, and a counterfeit coin is lighter than a genuine one. How can you determine 7 genuine coins in three weighings using a balance scale without weights?
1) Let's set aside one coin and put 31 coins on each pan of the scales. If the pans balance, then we have set aside the counterfeit coin, and there are 3 counterfeit coins on each pan. If one of the pans is heavier, then there are no more than three counterfeit coins on it. Thus, after the first weighing, we will be able to select 31 coins, among which there are no more than three counterfeit ones.
2) Take this group of coins and perform a similar operation: set aside one coin again and put 15 coins on each pan of the scales. After this weighing, we will be able to select 15 coins, among which there is no more than one counterfeit.
3) Repeating a similar operation for the third time, we will end up with 7 genuine coins.

At an international congress, 578 delegates from different countries arrived. Any three delegates can communicate with each other without the help of others (although one of them may have to translate the conversation between the other two). Prove that all delegates can be accommodated in double rooms of a hotel in such a way that any two living in the same room can communicate without external help.

## Solution

Take any three delegates; some two of them can definitely communicate with each other without a translator. We will house them in one room.

We will repeat this operation until only 4 delegates remain. As it is not difficult to verify by enumeration, they can always be divided into 2 rooms in such a way that those living in each room can communicate with each other without external help.