G5.1 If $1 \times 2 \times 3+2 \times 3 \times 4+3 \times 4 \times 5+\ldots+10 \times 11 \times 12=a$, find the value of $a$.
G5.2 Given that $5^{x}+5^{-x}=3$. If $5^{3 x}+5^{-3 x}=b$, find the value of $b$.
G5.3 Given that the roots of equation $x^{2}+m x+n=0$ are 98 and 99 and $y=x^{2}+m x+n$. If $x$ takes on the values of $0,1,2, \ldots, 100$, then there are $c$ values of $y$ that can be divisible by 6 . Find the value of $c$.

G5.4 In the figure, $A B C D$ is a square, $B F / / A C$, and $A E F C$ is a rhombus. If $\angle E A C=d^{\circ}$, find the value of $d$.
$$
\begin{array}{l} 
a=\frac{1}{4} n(n+1)(n+2)(n+3) \\
=\frac{1}{4} 10(11)(12)(13)=4290 \\
\left(5^{x}+5^{-x}\right)^{2}=9 \\
\Rightarrow 5^{2 x}+2+5^{-2 x}=9 \\
\Rightarrow 5^{2 x}+5^{-2 x}=7 \\
b=5^{3 x}+5^{-3 x} \\
=\left(5^{x}+5^{-x}\right)\left(5^{2 x}-1+5^{-2 x}\right) \\
=3(7-1)=18 \\
m=-98-99=-197 ; n=98 \times 99=49 \times 33 \times 6, \text { which is divisible by } 6 \\
y=x^{2}-197 x+98 \times 99 \\
=x^{2}+x-198 x+49 \times 33 \times 6 \\
=x(x+1)-6(33 x+49 \times 33)
\end{array}
$$

If $y$ is divisible by 6 , then $x(x+1)$ is divisible by 6
One of $x, x+1$ must be even. If it is divisible by 6 , then one of $x, x+1$ must be divisible by 3 .
We count the number of possible $x$ for which $y$ cannot be divisible by 6
These $x$ may be $1,4,7,10, \ldots, 97,100$; totally 34 possible $x$.
$$
c=101-34=67
$$

From $B$ and $E$ draw 2 lines $h, k \perp A C$
$$
h=k(\because B F / / A C)
$$

Let $A B=x, \angle C A B=45^{\circ}$
$$
k=x \sin 45^{\circ}=\frac{x}{\sqrt{2}}=h
$$
$$
\begin{array}{l}
A C=x \div \cos 45^{\circ} \\
=\sqrt{2} x=A E(\because A E F C \text { is a rhombus }) \\
\begin{aligned}
\sin \angle E A C & =\frac{h}{A E} \\
& =\frac{\frac{x}{\sqrt{2}}}{\sqrt{2} x}=\frac{1}{2}
\end{aligned}
\end{array}
$$
$$
d=30
$$