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Unorthodox Inequality

Let \(n\) be a positive integer and \(x\) be a positive real. Prove that \[\dfrac{n(nx+1)}{x+1} \le \dfrac{(x+1)^{2n}-1}{(x+1)^2-1}\] and find equality case.


Hint (zoom in to read):

\(\tiny _{\text{The inequality is still true even when loosening the restriction of }x\text{ to }x > -1}\)

Bigger hint: link

Note by Daniel Liu
3 years ago

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x>0 not x>=0 because if x=0 you get a non defined operation: 0/0 Nice problem ;) Carlos David Nexans · 3 years ago

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@Carlos David Nexans True. However, if you looked at the first hint, it can even be looser.

Has anyone solved this problem yet? I have a pretty big hint as my second hint... Daniel Liu · 3 years ago

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@Daniel Liu Nah I ended up at \(1 \geq 2\) and exploded the universe. Samuraiwarm Tsunayoshi · 2 years, 9 months ago

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Final hint: multiply \(x+1\) on both sides. Does the format of the resulting product resemble a known formula? Daniel Liu · 3 years ago

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