Question 12
(CZS 6) Given a segment $AB$ of the length $1$, define the set $M$ of points in the following way: it contains the two points $A,B$, and also all points obtained from $A,B$ by iterating the following rule: (*) for every pair of points $X,Y$ in $M$, the set $M$ also contains the point $Z$ of the segment $XY$ for which $YZ=3XZ$.
(a) Prove that the set $M$ consists of points $X$ from the segment $AB$ for which the distance from the point $A$ is either
$$AX=\frac{3k}{4^n}\quad\lor\quad AX=\frac{3k-2}{4^n},$$
where $n,k$ are nonnegative integers.
(b) Prove that the point $X_0$ for which $AX_0=1/2=X_0B$ does not belong to the set $M$.
(a) Prove that the set $M$ consists of points $X$ from the segment $AB$ for which the distance from the point $A$ is either
$$AX=\frac{3k}{4^n}\quad\lor\quad AX=\frac{3k-2}{4^n},$$
where $n,k$ are nonnegative integers.
(b) Prove that the point $X_0$ for which $AX_0=1/2=X_0B$ does not belong to the set $M$.