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## SEEMOUS 2012 Problem 1

Let $A=(a_{ij})$ be the $n\times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j+j^i$ by $3$ for $i,j=1,2,...,n$. Find the greatest $n$ for which $\det(A)\neq 0$.

The key is to observe that $i^j+j^i \equiv (i+6)^j+j^{i+6}$. Then if the matrix has more than $6$ lines, the determinant will be zero, since we have two equal lines. It now remains to calculate the determinants for $n\leq 6$.