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Algebra Difficulty 4.9 AIME Prove it Vietnam

Given a positive integer $n$ . Show that for any positive real number $x$ , we have the inequality:
$\frac{x^n (x^{n+1} + 1)}{x^n + 1} \le \left( \frac{x+1}{2} \right)^{2n+1}$
When does the equality take place?

Solution

The inequality is easily proved by induction on $n$ . The equality occurs if and only if $x = 1$ .

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