1981 USAMO Problems/Problem 5
Show that for any positive real ,
Solution(due to Pavel Zatitskiy)
First of all we write . So, we need to prove that Let's denote . It is easy to see that . We need to prove
We will prove it by induction by . The base is obvious, so we need to make a step.
Let's take such that is minimal. If then our inequality is obvious. So, . Then, by induction, and . Now we can add these two inequalities and get
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