Question 27
For which positive integers $n$ is the following statement true: if $a_{1}, a_{2}, \ldots, a_{n}$ are positive integers, $a_{k} \leq n$ for all $k$ and $\sum_{k=1}^{n} a_{k}=2 n$, then it is always possible to choose $a_{i_{1}}, a_{i_{2}}, \ldots, a_{i_{j}}$ in such a way that the indices $i_{1}, i_{2}, \ldots, i_{j}$ are different numbers, and $\sum_{k=1}^{j} a_{i_{k}}=n$?