G10.1 If $A$ is the area of a square inscribed in a circle of diameter 10 , find $A$.

G10.2 If $a+\frac{1}{a}=2$, and $S=a^{3}+\frac{1}{a^{3}}$, find $S$.
G10.3 An $n$-sided convex polygon has 14 diagonals. Find $n$.
G10.4 If $d$ is the distance between the 2 points $(2,3)$ and $(-1,7)$, find $d$.
Reference: 1985 FSG.4, 1989 FI3.3
Let the square be $B C D E$.
$$
\begin{array}{l}
B C=10 \cos 45^{\circ}=5 \sqrt{2} \\
A=(5 \sqrt{2})^{2}=50
\end{array}
$$

Reference: 1998 HG1
$$
\begin{aligned}
a^{2} & +\frac{1}{a^{2}}=\left(a+\frac{1}{a}\right)^{2}-2=4-2=2 \\
S & =a^{3}+\frac{1}{a^{3}} \\
& =\left(a+\frac{1}{a}\right)\left(a^{2}-1+\frac{1}{a^{2}}\right) \\
& =2(2-1)=2
\end{aligned}
$$

Reference: 1985 FG8.3, 1988 FG6.2, 1989 FG6.1, 1991 FI2.3, 2001 FI4.2, 2005 FI1.4
Number of diagonals $=C_{2}^{n}-n=\frac{n(n-1)}{2}-n=14$
$$
\begin{array}{l}
n^{2}-3 n-28=0 \\
(n-7)(n+4)=0 \\
\Rightarrow n=7
\end{array}
$$

Reference: 1986 FG9.4
$$
d=\sqrt{[2-(-1)]^{2}+(3-7)^{2}}=5
$$