Given natural $n$ $(n \ge 3)$ and positives $a_1, a_2, \cdots, a_{n – 1}, a_n$ such that $\displaystyle \prod_{i = 1}^na_i = 1$, prove that $$\large \prod_{i = 1}^n(a_i + 1)^{i + 1} > (n + 1)^{n + 1}$$
We have that $$\prod_{i = 1}^n(a_i + 1)^{i + 1} \ge \prod_{i = 1}^n(2\sqrt{a_i}) \cdot \left(\sqrt[m]{\prod_{i = 1}^na_i^i} + 1\right)^m$$
where $\displaystyle p = \sum_{i = 1}^ni = \dfrac{n(n + 1)}{2}$, then I don't know what to do next.
I suspect that $\displaystyle \min\left(\prod_{i = 1}^n(a_i + 1)^{i + 1}\right) = 2^q$, occuring when $a_1 = a_2 = \cdots = a_{n – 1} = a_n = 1$, where $q = \dfrac{(n + 3)n}{2}$, although I'm not sure that $2^q > (n + 1)^{n + 1}, \forall n \in \mathbb Z^+, n \ge 2$.
(I've just realised this is just a redraft of problem 2, IMO 2012.)
Best Answer
For the numbers $2^{\frac{n}{2}}$ is greater than n+1 obviously satisfies, for n>5 can be proven by induction (hint: x$\sqrt{2}$ - x - 1>0 for x> $\sqrt{2}$ + 1)and for other n<6 check manually