1998 IMO Shortlist Problems/A2
(Australia) Let be real numbers greater than or equal to 1. Prove that
Let denote the function .
Lemma 1. For , the function is decreasing.
Proof. Note that . Since is increasing for , the lemma follows.
Lemma 2. For positive , is convex.
Proof. Note that the derivative of is By Lemma 1, is increasing when , i.e., when . Therefore is convex for nonnegative .
For all integers , , so . Since is convex for nonnegative , it follows from Jensen's Inequality that as desired.
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