Problem: Find the last three digits of the product of the positive roots of $\sqrt{1995} x^{\log _{1995} x}=x^2$. Solution: Note that $LHS = x^{\frac{1}{2}\log_x{1995}+ \log_{1995}x}$ so taking $\log_x$ of both sides, we get $\frac{1}{2y}+y = 2$ where $y = \log_{1995}x$. The roots of this quadratic are $\frac{4\pm \sqrt{8}}{2} = 2\pm \sqrt{2}$. Therefore, we find that $x = 1995^{2 \pm \sqrt{2}}$, the product of these roots is $1995^2 \equiv 25 \pmod {1000}$ so the answer is $25$.