80. (40th IMO Problem) Let $n$ be a fixed integer, $n \geqslant 2$.
(1) Determine the smallest constant $c$ such that the inequality
$$\sum_{1 \leqslant i<j<n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{1 \leqslant i<n} x_{i}\right)^{4}$$

holds for all non-negative numbers $x_{1}, \cdots, x_{n} \geqslant 0$;
(2) For this constant $c$, determine the necessary and sufficient conditions for equality to hold.
80. Solution: (Using the local adjustment method) Since the inequality is homogeneous and symmetric, let $x_{1} \geqslant x_{2} \geqslant \cdots \geqslant x_{n}>0$, and $\sum_{i=1}^{n} x_{i}=1$. At this point, we only need to discuss
$$F\left(x_{1}, \cdots, x_{n}\right)=\sum_{1 \leqslant i<i \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right)$$

the maximum value.
Assume the last non-zero number among $x_{1}, \cdots, x_{n}$ is $x_{k+1}(k \geqslant 2)$, adjust $x=\left(x_{1}, \cdots, x_{k}, x_{k+1}, 0, \cdots, 0\right)$ to $x^{\prime}=\left(x_{1}, \cdots, x_{k-1}, x_{k}+x_{k+1}, 0, \cdots, 0\right)$, the corresponding function value
$$\begin{aligned}
F\left(x^{\prime}\right)-F(x)= & x_{k} x_{k+1}\left[3\left(x_{k}+x_{k+1}\right) \sum_{i=1}^{k-1} x_{i}-x_{k}^{2}-x_{k+1}^{2}\right]= \\
& x_{k} x_{k+1}\left[3\left(x_{k}+x_{k+1}\right)\left(1-x_{k}-x_{k+1}\right)-x_{k}^{2}-x_{k+1}^{2}\right]= \\
& x_{k} x_{k+1}\left\{\left(x_{k}+x_{k+1}\right)\left[3-4\left(x_{k}+x_{k+1}\right)\right]+2 x_{k} x_{k+1}\right\}
\end{aligned}$$

Since
so $\square$
Therefore
$$1 \geqslant x_{1}+x_{k}+x_{k+1} \geqslant \frac{1}{2}\left(x_{k}+x_{k+1}\right)+x_{k}+x_{k+1}$$

This shows that when $x$ is adjusted to $x^{\prime}$, the function value $F$ strictly increases. Thus, for any $x=\left(x_{1}, \cdots, x_{n}\right)$, after several adjustments, we get
$$\begin{aligned}
F(x) \leqslant & F(a, b, 0, \cdots, 0)=a b\left(a^{2}+b^{2}\right)= \\
& \frac{1}{2}(2 a b)(1-2 a b) \leqslant \frac{1}{8}= \\
& F\left(\frac{1}{2}, \frac{1}{2}, 0, \cdots, 0\right)
\end{aligned}$$

It is evident that the constant $c$ is $\frac{1}{8}$. The equality holds if and only if two $x_{i}$ are equal (which can be 0), and the remaining $n-2$ $x_{i}$ are all 0.