## PROBLEM 4

Let the points $A_{0}, A_{1}, A_{2}, \ldots, A_{n}$ be situated in this order on a line d such that $\mathrm{A}_{0} \mathrm{~A}_{1}=1 \mathrm{~cm}, \mathrm{~A}_{1} \mathrm{~A}_{2}=2 \mathrm{~cm}$, $\mathrm{A}_{2} \mathrm{~A}_{3}=2^{2} \mathrm{~cm}, \ldots, \mathrm{A}_{\mathrm{n}-1} \mathrm{~A}_{\mathrm{n}}=2^{\mathrm{n}-1} \mathrm{~cm}$.

a) Determine the natural number $\mathrm{p}$ such that $\mathrm{A}_{0} \mathrm{~A}_{\mathrm{p}}=2047 \mathrm{~cm}$.

b) If $\mathrm{M}$ is the midpoint of the segment $\mathrm{A}_{2} \mathrm{~A}_{12}$ and $\mathrm{N}$ is the midpoint of the segment $\mathrm{A}_{4} \mathrm{~A}_{10}$, determine the length of the segment MN.

(Mathematical Gazette)
## Solution:

a) $\mathrm{A}_{0} \mathrm{~A}_{\mathrm{p}}=\mathrm{A}_{0} \mathrm{~A}_{1}+\mathrm{A}_{1} \mathrm{~A}_{2}+\ldots+\mathrm{A}_{\mathrm{p}-1} \mathrm{~A}_{\mathrm{p}} \Rightarrow \mathrm{A}_{0} \mathrm{~A}_{\mathrm{p}}=1+2+2^{2}+\ldots+2^{\mathrm{p}-1}=2^{\mathrm{p}}-1$
$1 \mathrm{p}$

$2^{\mathrm{p}}-1=2047 \Rightarrow 2^{\mathrm{p}}=2^{11} \Rightarrow \mathrm{p}=11$
b) $\mathrm{M}$ is the midpoint of segment $\mathrm{A}_{2} \mathrm{~A}_{12} \Rightarrow \mathrm{A}_{2} \mathrm{M}=\frac{\mathrm{A}_{2} \mathrm{~A}_{12}}{2} \Rightarrow \mathrm{A}_{2} \mathrm{M}=2046 \mathrm{~cm}$ and $\mathrm{A}_{0} \mathrm{M}=2049 \mathrm{~cm}$

$\mathrm{N}$ is the midpoint of segment $\mathrm{A}_{4} \mathrm{~A}_{10} \Rightarrow \mathrm{A}_{4} \mathrm{~N}=\frac{\mathrm{A}_{4} \mathrm{~A}_{10}}{2} \Rightarrow \mathrm{A}_{4} \mathrm{~N}=504 \mathrm{~cm}$ and $\mathrm{A}_{0} \mathrm{~N}=519 \mathrm{~cm}$