Problem 4. A set will be called interesting if its elements are prime numbers and the following condition is met:

no matter how we choose three distinct elements of the set, the sum of the chosen numbers is a prime number.

Determine the maximum number of elements an interesting set can have.
Solution. The answer is four $1 p$

Let's show that from any set of five numbers, we can choose three such that their sum is divisible by 3.

Indeed, if three of the chosen numbers give the same remainder when divided by 3, then their sum is divisible by 3. Otherwise, for each remainder when divided by 3, we have at most two numbers among the chosen ones, so we have at least one number that gives a remainder of 0, one number that gives a remainder of 1, and one number that gives a remainder of 2; the sum of these numbers is divisible by 3.

From the above, it follows that an interesting set cannot have five or more elements, because in this case we find three numbers whose sum is divisible by 3, and their sum is greater than 3. An interesting set of four numbers is $7, 13, 17, 23 \ldots 3 \mathbf{p}$