G4.1 If $a$ is the sum of all the positive factors of 2002, find the value of $a$.
G4.2 It is given that $x>0, y>0$ and $\sqrt{x}(\sqrt{x}+\sqrt{y})=3 \sqrt{y}(\sqrt{x}+5 \sqrt{y})$. If $b=\frac{2 x+\sqrt{x y}+3 y}{x+\sqrt{x y}-y}$, find the value of $b$.

G4.3 Given that the equation ||$x-2|-1|=c$ has only 3 integral solutions, find the value of $c$.
G4.4 If $d$ is the positive real root of the equation $\frac{1}{2}\left\{\frac{1}{2}\left[\frac{1}{2}\left(\frac{1}{2} x^{2}+2\right)+2\right]+2\right\}=2$, find the value of $d$.
Reference 1993 HI8, 1994 FI3.2, 1997 HI3, 1998 HI10, 1998 FI1.4, 2005 FI4.4
$$
2002=2 \times 7 \times 11 \times 13
$$

The positive factors may be $2^{a} 7^{b} 11^{c} 13^{d}$, where $0 \leq a, b, c, d \leq 1$ are integers.
The sum of all positive factors are $(1+2)(1+7)(1+11)(1+13)=3 \times 8 \times 12 \times 14=4032=a$
$$
\begin{array}{l}
\sqrt{x}(\sqrt{x}+\sqrt{y})=3 \sqrt{y}(\sqrt{x}+5 \sqrt{y}) \\
\Rightarrow x+\sqrt{x y}=3 \sqrt{x y}+15 y \\
\Rightarrow x-2 \sqrt{x y}-15 y=0 \\
(\sqrt{x}+3 \sqrt{y})(\sqrt{x}-5 \sqrt{y})=0 \\
\Rightarrow \sqrt{x}=5 \sqrt{y} \text { and } x=25 y
\end{array}
$$
$$
\begin{array}{l}
b=\frac{2 x+\sqrt{x y}+3 y}{x+\sqrt{x y}-y}=\frac{50 y+\sqrt{25 y^{2}}+3 y}{25 y+\sqrt{25 y^{2}}-y}=\frac{58 y}{29 y}=2 \\
b=2
\end{array}
$$

Reference: 2005 FG4.2, 2009 HG9, 2012 FG4.2
$$
\begin{array}{l}
|x-2|-1= \pm c \\
\Rightarrow|x-2|=1 \pm c
\end{array}
$$

In order that it has only 3 integral solutions
$$
\begin{array}{l}
c=1 \\
\frac{1}{2}\left\{\frac{1}{2}\left[\frac{1}{2}\left(\frac{1}{2} x^{2}+2\right)+2\right]+2\right\}=2 \\
\Rightarrow\left\{\frac{1}{2}\left[\frac{1}{2}\left(\frac{1}{2} x^{2}+2\right)+2\right]+2\right\}=4 \\
\Rightarrow \frac{1}{2}\left[\frac{1}{2}\left(\frac{1}{2} x^{2}+2\right)+2\right]=2 \\
{\left[\frac{1}{2}\left(\frac{1}{2} x^{2}+2\right)+2\right]=4} \\
\Rightarrow \frac{1}{2}\left(\frac{1}{2} x^{2}+2\right)=2 \\
\Rightarrow \frac{1}{2} x^{2}+2=4 \\
\Rightarrow \frac{1}{2} x^{2}=2 \\
\Rightarrow x^{2}=4 \\
\Rightarrow x= \pm 2 \\
d=\text { the positive real root }=2
\end{array}
$$