27. (POL 2) Determine the maximum value of the sum
$$
\sum_{i<j} x_{i} x_{j}\left(x_{i}+x_{j}\right)
$$
over all $n$-tuples $\left(x_{1}, \ldots, x_{n}\right)$, satisfying $x_{i} \geq 0$ and $\sum_{i=1}^{n} x_{i}=1$.
27. Write $F\left(x_{1}, \ldots, x_{n}\right)=\sum_{i0,
\end{array}
$$
because $x_{k-1}+x_{k} \leq \frac{2}{3}\left(x_{1}+x_{k-1}+x_{k-2}\right) \leq \frac{2}{3}$. Repeating this procedure we can reduce the number of nonzero $x_{1}$ 's to two, increasing the value of $F$ in each step. It remains to maximize $F$ over $n$-tuples $\left(x_{1}, x_{2}, 0, \ldots, 0\right)$ with $x_{1}, x_{2} \geq 0, x_{1}+x_{2}=1$ : in this case $F$ equals $x_{1} x_{2}$ and attains its maximum value $\frac{1}{4}$ when $x_{1}=x_{2}=\frac{1}{2}, x_{3}=\ldots, x_{n}=0$