Saudi Arabian Mathematical Competitions

Assume that $1-5^{n}+5^{2 n+1}=m^{2}$, for a positive integer $m$. We have
$$5^{n}\left(5^{n+1}-1\right)=(m-1)(m+1) .$$
Because $(m+1)-(m-1)=2$, the number $5$ cannot divide both $m-1$ and $m+1$. Therefore, we have two cases:

First case when $5$ divides $m-1$. In this case, there exists a positive integer $k$ such that $m-1=5^{n} k$ and $5^{n+1}-1=(m+1) k=\left(5^{n} k+2\right) k$. We, therefore, obtain the equation
$$5^{n}\left(5-k^{2}\right)=2 k+1$$
For $k=1$, this equation has no solution. For $k=2$, $n=1$ is a solution for the equation. For $k \geq 3$ this equation has no solution since its left hand side is negative.

Second case when $5$ divides $m+1$. In this case, there exists a positive integer $k$ such that $m+1=5^{n} k$ and $5^{n+1}-1=(m-1) k=\left(5^{n} k-2\right) k$. We, therefore, obtain the equation
$$5^{n}\left(k^{2}-5\right)=2 k-1$$
This equation has no solution since for $k=1,2$ or $3$ the left hand side is either negative or even, and for $k \geq 4$ we have
$$5^{n}\left(k^{2}-5\right)>5(k+2)(k-3) \geq 5 k+10>2 k-1 .$$
Therefore, the unique solution to this problem is $n=1$ since we have
$$1-5+5^{3}=11^{2} .$$