## SUBJECT III (7p)

Let $\mathrm{A}$ and $\mathrm{B}$ be two points in the plane such that $\mathrm{AB}=1 \mathrm{~m}, \mathrm{M}_{1}$ is the midpoint of the segment $[\mathrm{AB}], \mathrm{M}_{2}$ is the midpoint of the segment [AM $M_{1}$ ], $M_{3}$ is the midpoint of the segment $\left[A M_{2}\right.$ ], and so on, $M_{n}$ is the midpoint of the segment $\left[A M_{n-1}\right]$ for any natural number $n$.

a) Calculate the length of the segment $\left[\mathbf{M}_{5} \mathbf{M}_{4}\right]$.

b) Determine the smallest number $\mathbf{n}$ for which the length of the segment $\left[A M_{n}\right]$ is less than $1 \mathrm{~mm}$.
## Subject III

![](https://cdn.mathpix.com/cropped/2024_06_07_3c25989a4f125ee3747ag-3.jpg?height=143&width=1941&top_left_y=471&top_left_x=66)
a) $A B=1 \mathrm{~m}=1000 \mathrm{~mm}$

$A M_{1}=A B: 2=500 \mathrm{~mm}$

$A M_{2}=A M_{1}: 2=A B: 2^{2}=250 \mathrm{~mm}, A M_{3}=A M_{2}: 2=A B: 2^{3}=125 \mathrm{~mm}$

$A M_{4}=A M_{3}: 2=A B: 2^{4}=62.5 \mathrm{~mm}, A M_{5}=A M_{4}: 2=A B: 2^{5}=31.25 \mathrm{~mm}$

$\mathrm{M}_{5} \mathrm{M}_{4}=A M_{5}=31.25 \mathrm{~mm}$

$2 p$

b) It is observed that $A M_{n}=A B: 2^{n}$ $2 p$

$\Rightarrow A M_{n}A B, A B=1000 \mathrm{~mm}$ $1 p$

$2^{10}=1024$

$2^{9}=512 \quad \Rightarrow n=10$ is the smallest value.