Solution
= Solution
The <cumulative distribution function> is <continuous> and <strictly increasing> for $x>0$, so the <median> is the unique positive $c$ satisfying $F_X(c)=1/2$. Consequently $e^{-c^2/2}=1/2$, giving
$$
\boxed{c=\sqrt{2\log2}.}
$$
There is no atom at $c$, so $\mathbb P(X>c)=1-F_X(c)=1/2$ as required.