2. Given positive numbers $a, b, c, d$. Find the minimum value of the expression

$$
A=\left(a^{2}+\frac{1}{b c}\right)^{3}+\left(b^{2}+\frac{1}{c d}\right)^{3}+\left(c^{2}+\frac{1}{d a}\right)^{3}+\left(d^{2}+\frac{1}{a b}\right)^{3}
$$
Answer: 32.

Solution. We will use the Cauchy inequality for means first in each parenthesis, and then for the entire sum. We get

$$
A \geqslant\left(\frac{2 a}{\sqrt{b c}}\right)^{3}+\left(\frac{2 b}{\sqrt{c d}}\right)^{3}+\left(\frac{2 c}{\sqrt{d a}}\right)^{3}+\left(\frac{2 d}{\sqrt{a b}}\right)^{3} \geqslant 32\left(\frac{a^{2}}{b c} \cdot \frac{b^{2}}{c d} \cdot \frac{c^{2}}{d a} \cdot \frac{d^{2}}{a b}\right)^{3 / 8}=32
$$

Equality is achieved when $a=b=c=d=1$.