Vijetnam, 2011

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 $x = 1$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.