Polansky $A$.

Let $p$ be a prime number. A set of $p+2$ natural numbers (not necessarily distinct) is called interesting if the sum of any $p$ of them is divisible by each of the two remaining numbers. Find all interesting sets.
Let $S$ be the sum of all numbers in an interesting set, $c$ be the largest number, and $a$ and $b$ be any two other numbers in the set. The sums $S-a-c$ and $S-b-c$ are divisible by $c$, so their difference $b-a$ is also divisible by $c$. Since this difference in absolute value is less than $c$, it must be zero. Therefore, all numbers in the set, except for the largest number $c$, are equal to $a$.

$S-a-a=(p-1) a+c$ is divisible by $a$, hence $c=k a$.

$S-a-c=p a$ is divisible by $c$, which is equivalent to the divisibility of $p$ by $k$. Therefore, $k=1$ or $k=p$.

## Answer

Sets of the forms $(p m, m, \ldots, m)$ and $(m, m, \ldots, m)$, where $m$ is a natural number.