Problem: Let $a_2$, $a_3$, \dots, $a_n$ be positive reals with product $1$,
where $n \ge 3$.
Show that
\[ (1+a_2)^2 (1+a_3)^3 \dots (1+a_n)^n > n^n. \]

Solution: Try the dumbest thing possible: by AM-GM,
\begin{align*}
  (1 + a_2)^2 &\ge 2^2 a_2 \\
  (1 + a_3)^3 = \left( \half + \half + a_3 \right)^3 &\ge \frac{3^3}{2^2} a_3 \\
  (1 + a_4)^4 = \left( \frac13 + \frac13 + \frac13 + a_4 \right)^4
    &\ge \frac{4^4}{3^3} a_4 \\
  &\vdotswithin=
\end{align*}
and so on.
Multiplying these all gives the result.
The inequality is strict since it's not possible
that $a_2 = 1$, $a_3 = \half$, et cetera.