SCHWEITZER 1949 Problem 4

Let $A$ and $B$ be two disjoint sets in the interval $(0,1)$ . Denoting by $\mu$ the Lebesgue measure on the real line, let $\mu(A)>0$ and $\mu(B)>0$ . Let further $n$ be a positive integer and $\lambda =\frac1n$ . Show that there exists a subinterval $(c,d)$ of $(0,1)$ for which $\mu(A\cap (c,d))=\lambda \mu(A)$ and $\mu(B\cap (c,d))=\lambda \mu(B)$ . Show further that this is not true if $\lambda$ is not of the form $\frac1n$ .