Thinking Question 2 (40th IMO Problem) Let $n$ be a fixed integer, $n \geqslant 2$.
a) Determine the smallest constant $c$ such that the inequality $\sum_{1 \leqslant i<1 \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{i=1}^{n} x_{i}\right)^{4}$

holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n} \geqslant 0$;
b) For this constant $c$, determine the necessary and sufficient conditions for equality to hold.
a) When non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$ are not all 0, let
$$
x=\frac{\sum x_{i} x_{j}}{\left(\sum_{i=1}^{n} x_{i}\right)^{2}}, y=\frac{\sum x_{i} x_{j} x_{k}\left(x_{i}+x_{j}+x_{k}\right)}{\left(\sum_{i=1}^{n} x_{i}\right)^{\prime}} .
$$

Since
$$
\begin{aligned}
& \sum x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right)=\sum x_{i} x_{j}\left[\left(\sum_{k=1}^{n} x_{k}\right)^{2}-2 \sum x_{i} x_{j}-\sum_{\substack{k=1 \\
(k \neq i, j)}}^{n} x_{k}\left(x_{i}+x_{j}+x_{k}\right)\right] \\
= & \sum x_{i} x_{j}\left(\sum_{k=1}^{n} x_{k}\right)^{2}-2\left(\sum x_{i} x_{j}\right)^{2}-\sum x_{i} x_{j} x_{k}\left(x_{i}+x_{j}+x_{k}\right),
\end{aligned}
$$

thus equation (1) $\Leftrightarrow c \geqslant-2 x^{2}+x-y$.
Since $-2 x^{2}+x-y \leqslant \frac{1}{8}$, the equality holds if and only if $x=\frac{1}{4}$ and $y=0$,
thus $c \geqslant \frac{1}{8}, c_{\text {min }}=\frac{1}{8}$, which also holds when $x_{1}=x_{2}=\cdots=x_{n}=0$.
b) When $c=\frac{1}{8}$, $x=\frac{1}{4}$ and $y=0$ if and only if
$$
\left(\sum_{i=1}^{n} x_{i}\right)^{2}=4 \sum x_{i} x_{j} \text { and } \sum x_{i} x_{j} x_{k}\left(x_{i}+x_{j}+x_{k}\right)=0 \Leftrightarrow \sum_{i=1}^{n} x_{i}^{2}=2 \sum x_{i} x_{j} \text { and } \sum x_{i} x_{j} x_{k}=0 \text {. }
$$

Since $\sum x_{i} x_{j} x_{k}=0 \Leftrightarrow x_{1}, x_{2}, \cdots, x_{n}$ have at most two non-zero terms $x_{i} 、 x_{j}$, and at this time $\sum_{i=1}^{n} x_{i}^{2}=2 \sum x_{i} x_{j} \Leftrightarrow x_{i}^{2}+x_{j}^{2}=2 x_{i} x_{j} \Leftrightarrow x_{i}=x_{j}$.

Therefore, $x=\frac{1}{4}$ and $y=0$ if and only if $x_{1}, x_{2}, \cdots, x_{n}$ have two equal terms (which can be 0), and the rest are all 0.