2. Determine for which natural numbers $n$ ( $n \geqq 2$ ) it is possible to select at least two distinct even numbers from the set $\{1,2, \ldots, n-1\}$ such that their sum is divisible by the number $n$.
SOLUTION. If $n$ is even and the set contains even numbers 2 and $n-2$, then 24 satisfy the condition of the problem.

From the sets $\{1\}$ (for $n=2$) and $\{1,2,3\}$ (for $n=4$), the required selection cannot obviously be made.

If $n$ is odd, we can analogously select three even numbers 4, $n-3, n-1$ from the given sets for $n>7$, where $4<n-3<n-1$, with the sum $4+(n-3)+(n-1)=2n$, which is divisible by $n$.

From the sets $\{1,2\}$ (for $n=3$), $\{1,2,3,4\}$ (for $n=5$), and $\{1,2,3,4,5,6\}$ (for $n=7$), it is clearly not possible to select either two or three different even numbers $s$ with the required property.

The problem's condition is satisfied by the number $n=6$ and all natural numbers $n \geq 8$.

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