Question 6
Let $n \ge 2$ and consider a table with $n+1$ rows and $n$ columns, in which, on each of the first $n$ rows, Nicuşor writes, in some order, the numbers $1, 2, \dots, n$. Then, he chooses a permutation $a_1, a_2, \dots, a_n$ of the numbers $1, 2, \dots, n$ and completes the last row as follows: for each $j \in \{1, 2, \dots, n\}$, he writes in the cell at column $j$ the number of occurrences of $a_j$ in the cells located at the intersection of column $j$ with the first $n$ rows.
Determine all $n$ for which Nicuşor can complete the table and choose $a_1, a_2, \dots, a_n$ so that the last row contains, in some order, the numbers $1, 2, \dots, n$.
Cristi Săvescu
Determine all $n$ for which Nicuşor can complete the table and choose $a_1, a_2, \dots, a_n$ so that the last row contains, in some order, the numbers $1, 2, \dots, n$.
Cristi Săvescu