a³ + b³ + c³ ≥ a²b + b²c + c²a

Rearrangement. The sequences $(a,b,c)$ and $(a^2,b^2,c^2)$ are ordered in the same way. The sum of products of the identically ordered terms is at least the sum for any other pairing:
$$a\cdot a^2+b\cdot b^2+c\cdot c^2\ge b\cdot a^2+c\cdot b^2+a\cdot c^2=a^2b+b^2c+c^2a.$$

AM-GM. $\frac{a^3+a^3+b^3}3\ge\sqrt[3]{a^6b^3}=a^2b$, i.e. $2a^3+b^3\ge3a^2b$. Similarly $2b^3+c^3\ge3b^2c$ and $2c^3+a^3\ge3c^2a$. Adding: $3(a^3+b^3+c^3)\ge3(a^2b+b^2c+c^2a)$.

Equality iff $a=b=c$.

Lesson. Rearrangement is the tool when two sequences are *similarly ordered*; AM-GM with repeated terms is the tool when you need to produce a specific monomial like $a^2b$.