Problem 962

Solution. We will not perform redistributions by columns. First, from the second column, we move to the first column every competitor who does not have an acquaintance in the first column. The new arrangement is good, so at least one competitor remains in the second column. Now, from the third column, we move to the second column every competitor who does not have an acquaintance among the remaining in the second column. The new arrangement is again good, so at least one competitor remains in the third column, etc. Finally, from the $n$-th column, we move to the $(n-1)$-th column every competitor who does not have an acquaintance among the remaining in the $(n-1)$-th column. The obtained arrangement is again good, so at least one competitor remains in the $n$-th column. We denote one such by $M_{n}$. There must be an acquaintance $M_{n-1}$ of $M_{n}$ in the $(n-1)$-th column. Let us note that $M_{n-1}$ has not been moved (otherwise, the initial arrangement would not be good). Therefore, $M_{n-1}$ has an acquaintance $M_{n-2}$ in the $(n-2)$-th column. Analogously, we conclude that $M_{n-2}$ has not been moved, etc. In this way, competitors $M_{1}, M_{2}, \ldots, M_{n}$ are found for whom conditions (1) and (2) hold.