## SUBJECT 3.

$O, A, B, C$ are collinear points in this order, $O A=2^{x} \mathrm{~cm}, O B=2^{x+1} \mathrm{~cm}, O C=2^{x+2} \mathrm{~cm}, x \in \mathbf{N}^{*}$.

a) Show that $B C=O A+A B$;

b) If $M$ is the midpoint of segment $[O A]$ and $N$ is the midpoint of segment $[B C]$, and $M N=20 \mathrm{~cm}$, find the length of segment $[A C]$.
## Subject 3.

a) $O A+A B=O B=2^{x+1}(\text{or } O B=2 \cdot O A, O C=4 \cdot O A)$ $1 p$

$\mathrm{BC}=\mathrm{OC}-\mathrm{OB}=2^{\mathrm{x}+2}-2^{\mathrm{x}+1}=2^{\mathrm{x}+1}=\mathrm{OA}+\mathrm{AB}(\mathrm{BC}=\mathrm{OB}=\mathrm{OA}+\mathrm{AB})$

$1 p$
a) $M N=2^{x+1}+2^{x-1}\left(\text{or } M N=\frac{5}{2} \cdot O A\right)$

$2 p$

$x=3 \mathrm{~cm}$

$2 p$

$A C=24 \mathrm{~cm}$

$1 p$