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I've encountered the following problem that I don't know how to solve:
Given positive natural $n$ and positive real $x_1, x_2, ..., x_n$ prove that there exists such positive natural $N$ that
$(1+\frac1n)^N\ge 2 (x_1+x_2+...+x_{n-1}) + \frac{N}nx_n\ge N^n$.
I don't even know how to start with either of the parts, leave alone both of them simultaneously. I supposed it may be done by the inequality of means but my attempts didn't show anything interesting.
Appreciate your help.

Sorry Norry
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  • @EricTowers x. My bad – Sorry Norry Dec 05 '18 at 20:04
  • If $n=2$ and $x_1=x_2=1$ the inequality reduces to $(3/2)^N\geq N-1\geq N^2$, and the second inequality can't be true for any $N$. Do you have a typo somewhere? – Joel Moreira Dec 05 '18 at 20:33
  • @JoelMoreira Thank you. I made an error. The correct “developed” fraction in the middle should be reducible to 2. I was too careless. I know it changes a lot. Could you please help me nonetheless? – Sorry Norry Dec 05 '18 at 20:50
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    Hm, still when $n=2$ and $x_1=x_2=1/4$ the second inequality becomes $1/2+N/8 \geq N^2$ which does not have a solution $N$. – Joel Moreira Dec 05 '18 at 20:59
  • More generally, the middle term is linear in $N$ and the right hand side is quadratic when $n=2$, so the second inequality will probably have no solution, unless something changes more dramatically. – Joel Moreira Dec 05 '18 at 21:01

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