With abc = 1: Σ 1/(a² + a + 1) ≥ 1

Idea. The constraint kills most terms after expansion.

Clearing denominators. Let $D=\prod(a^2+a+1)$. The inequality is
$$\sum_{\rm cyc}(b^2+b+1)(c^2+c+1)-D\ge0.$$

Expansion. Expanding mechanically gives a polynomial whose terms, after substituting $abc=1$, simplify:
- $-a^2b^2c^2=-1$,
- $-a^2b^2c-a^2bc^2-ab^2c^2=-ab-ac-bc$,
- $-a^2bc-ab^2c-abc^2=-a-b-c$,
- $-abc=-1$.

Combining with the positive terms $a^2+b^2+c^2+a+b+c+2$:
$$a^2+b^2+c^2-ab-bc-ca\ \ge0.$$

Equality. Iff $a=b=c=1$.

Lesson. With a multiplicative constraint, expand fully and use $abc=1$ to cancel: often the inequality collapses to a classical one such as $a^2+b^2+c^2\ge ab+bc+ca$.