Assume that $n \in \mathbb{N}, n \geq 2$ and $a_i \in \mathbb{R}, a_i > 0, \forall i=1,...,n$
Show that $$ \frac{a_1 (\sum_{i=2}^n a_i^{n-1})}{ a_1^n + \sum_{i=2}^n a_i^n} \leq 1 -\frac{1}{n}$$
Progress. Let $\bar{a} = \sum_{i=2}^n a_i^{n-1}/(n-1)$, by using power mean inequality, the LHS becomes $$ \frac{a_1 (n-1) \bar{a}}{a_1^n + (n-1)\bar{a}^{\frac{n}{n-1}}}$$ But I don't know how to proceed?
According to the AM-GM Inequality on $n$ variables: $$a_1^n+(n-1)a_i^n=a_1^n+[a_i^n+a_i^n+\cdots +a_i^n]\ge na_1a_i^{n-1}$$ Using, this, we see that: $$\begin{align*} n\sum_{i=2}^na_1a_i^{n-1} &\le \sum_{i=2}^{n-1} [a_1^n+(n-1)a_i^n] \\ &=(n-1)\left[a_1^n+\sum_{i=2}^na_i^n\right] \end{align*}$$ And this rearranges to the desired.