How should we choose the positive numbers $a, b, c$; if their sum is constant, so that the expression $a b^{2} c^{3}$ is as large as possible?
$$
\begin{aligned}
a b^{2} c^{3} & =\left(a \frac{1}{6} \frac{1}{3} \frac{1}{2}\right)^{6}=\frac{1}{6 \cdot 3^{2} \cdot 2^{3}}\left((6 a)^{\frac{1}{6}(3 b)^{\frac{1}{3}}(2 c) \frac{1}{2}}\right)^{6} \leqq \\
& \leqq \frac{1}{2^{4} \cdot 3^{3}}\left(\frac{1}{6}(6 a)+\frac{1}{3}(3 b)+\frac{1}{2}(2 c)\right)^{6}=\frac{A^{6}}{2^{4} \cdot 3^{3}}
\end{aligned}
$$

Where $A$ is the sum of the numbers. The expression in question will be the largest when $6 a=3 b=2 c$, that is,

$$
a=\frac{A}{6}, \quad b=2 a=\frac{A}{3}, \quad c=3 a=\frac{A}{2}
$$