There are three arbitrary random variables on $\mathbb{R}$: $X$, $Y$, $Z$. The median is defined as $\mathrm{Me}(X) = \mathrm{sup} \left\{t: F_X(t) \le \frac12\right\}$ (so it's hopefully always unique). We know that, say, $\mathrm{Me}(X) \geq \mathrm{Me}(Y) \geq \mathrm{Me}(Z)$. Is it true that it always exists $\alpha$ such that $\mathrm{Me}(Y) = \mathrm{Me} (\alpha X + (1-\alpha) Z)$?
My conjecture is that $f(\alpha) = \mathrm{Me} (\alpha X + (1-\alpha) Z)$ should be continuous, hence the claim above is true, but I can't prove it formally or find appropriate reference.