Consider iid samples from a fixed distribution function $F(x)$ and consider its median. Now consider another median from iid samples where one half is drawn from $F_1(x)$ and the other is drawn from $F_2(x)$ with $F_1(x)+F_2(x)=2F(x)$ for all $x$. Which median has a larger variance?
I did a Google search for variance of the sample median and found the 1978 paper by Maritz and Jarrett. I remember this paper as it is referenced in my bootstrap book where I used it to show an example where the bootstrap distribution can be obtained without need for the Monte Carlo approximation. I didn't go to the trouble of working out the solution to this problem based on the paper because this is a homework type problem and the site suggests providing hints to the solution rather than doing it for the author of the question. The paper is in JASA 1978.