(British MO) Determine the greatest value that the expression

$$
x^{2} y+y^{2} z+z^{2} x-x y^{2}-y z^{2}-z x^{2}
$$

can take for $0 \leqslant x, y, z \leqslant 1$.
Let $S(x, y, z)$ be the expression to maximize.

Note that the expression is homogeneous and the maximum is positive. Thus, if $x, y, z \leqslant 0$, then for $k>1$ such that $k x, k y, k z \leqslant 1$, we have $S(k x, k y, k z)>S(x, y, z)$. We deduce that the maximum value of $S$ is achieved when one of the variables equals 1 (otherwise we have shown that we can find another triplet achieving a higher value). Without loss of generality, we can assume that $x=1$. We then need to maximize

$$
y+y^{2} z+z^{2}-y^{2}-y z^{2}-z=(1-y)(1-z)(y-z)
$$

By the AM-GM inequality, this expression is less than or equal to

$$
(1-z)\left(\frac{y-z+1-z}{2}\right)^{2}=\frac{(1-z)^{3}}{4} \leqslant \frac{1}{4}
$$

Equality is achieved for $y=1 / 2$ and $z=0$.