G4.1 If $0 . \dot{1}+0.0 \dot{2}+0.00 \dot{3}+\ldots+0.00000000 \dot{9}=a$, find the value of $a$ (Give your answer in decimal)
G4.2 The circle in the figure has centre $O$ and radius $1, A$ and $B$ are points on the circle. Given that $\frac{\text { Area of shaded part }}{\text { Area of unshaded part }}=\frac{\pi-2}{3 \pi+2}$ and $\angle A O B=b^{\circ}$, find the value of $b$.

G4.3 A sequence of figures $S_{0}, S_{1}, S_{2}, \ldots$ are constructed as follows. $S_{0}$ is obtained by removing the middle third of $[0,1]$ interval; $S_{1}$ by removing the middle third of each of the two intervals in $S_{0} ; S_{2}$ by removing the middle third of each of the four intervals in $S_{1} ; S_{3}, S_{4}, \ldots$ are obtained similarly. Find the total length $c$ of the intervals removed in the construction of $\mathrm{S}_{5}$ (Give your answer in fraction).

G4.4 All integers are coded as shown in the following table. If the sum of all integers coded from 101 to 200 is $d$, find the value of $d$.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline Integer & $\ldots$ & $\ldots$ & -3 & -2 & -1 & 0 & 1 & 2 & 3 & $\ldots$ & $\ldots$ \\
\hline Code & $\ldots$ & $\ldots$ & 7 & 5 & 3 & 1 & 2 & 4 & 6 & $\ldots$ & $\ldots$ \\
\hline
\end{tabular}
$$
\begin{array}{l}
a=\frac{1}{9}+\frac{2}{90}+\frac{3}{900}+\cdots+\frac{9}{900000000}=\frac{100000000+20000000+3000000+\cdots+9}{900000000} \\
a=\frac{123456789}{900000000}=\frac{13717421}{100000000}=0.13717421 \\
\frac{\text { Area of shaded part }}{\text { Area of the circle }}=\frac{\pi-2}{\pi-2+3 \pi+2}=\frac{\pi-2}{4 \pi} \\
\frac{\pi(1)^{2} \cdot \frac{b}{360}-\frac{1}{2}(1)^{2} \sin b^{\circ}}{\pi(1)^{2}}=\frac{\pi-2}{4 \pi} \Rightarrow \frac{\pi b}{90}-2 \sin b^{\circ}=\pi-2 ; b=90 \\
\text { L }) \longrightarrow \mathrm{S}_{0} \\
0 \quad \frac{1}{3} \quad \frac{2}{3} \quad 1 \\
1=(\quad) \longrightarrow(\quad)-1 S_{1} \\
\begin{array}{llllllll}
0 & \frac{1}{9} & \frac{2}{9} & \frac{1}{3} & \frac{2}{3} & \frac{7}{9} & \frac{8}{9} & 1
\end{array} \\
\text { |-( )-( )-( )-( )-( )-( )-( )-| S S } \\
\end{array}
$$

The total length in $S_{0}=\frac{2}{3}$
The total length in $S_{1}=4 \times \frac{1}{9}=\frac{4}{9}$
The total length in $S_{2}=8 \times \frac{1}{27}=\frac{8}{27}$
Deductively, the total length in $S_{5}=2^{6} \times \frac{1}{3^{6}}=\frac{64}{729}$
The total length removed in $S_{5}=1-\frac{64}{729}=\frac{665}{729}$
Sum of integers code as $102,104, \ldots, 200$ is $51+52+\ldots+100$
Sum of integers code as $101,103, \ldots, 199$ is $-50-51-\ldots-99$
Sum of all integers $=1+1+\ldots+1(50$ times $)=50$