Solution
= Solution
The distribution function is $F(x)=x^2/\theta^2$ on $[0,\theta]$, so its median is $\theta/\sqrt2$. The <likelihood function> is
$$
L(\theta)=2^n\left(\prod_iX_i\right)\theta^{-2n}\mathbf1_{\{\theta\geq X_{(n)}\}},
$$
which decreases over its admissible range. Hence $\widehat\theta=X_{(n)}=\max_iX_i$. By the <invariance property of maximum likelihood estimation>,
$$
\boxed{\widehat m=\frac{X_{(n)}}{\sqrt2}.}
$$