Regional Competition

We immediately see that $n=1$ does not lead to a solution, while $n=2$ yields the solution $(k,n)=(45,2)$.

We show that there is no solution with $n\ge 3$. In that case $3^n$ is divisible by $9$ and thus $k^2$ is divisible by $9$ which implies that $k=3l$ for some positive integer $l$. After division by $9$, the equation reads $l^2-224=3^{n-2}$. Modulo $3$, this yields $l^2-2\equiv 0(\mathrm{mod}3)$, a contradiction because $2$ is not a quadratic residue modulo $3$.