avatarWojciech kowalczyk


An inequality from the Junior Balkan Mathematical Olympiad

This inequality was asked to young teenagers from the Balkans, but this does not make this problem any easier! These teenagers were at most 15.5 years old but were of the highest calibre that Europe could offer. Before looking at the solution, give the problem a try, it may not be as easy as you think!

Before we look at the solution, we have to introduce some tools that help us along the way. First, we introduce cyclic sums. We have the following:

In other words, if we sum something cyclically, we rotate between each of the variables. Here is an example. Suppose we had 3 variables a,b,c. Then:

How does this relate to our problem? Well, this is equivalent to writing the following:

This may not seem helpful at first glance, but this is where we include a tool that will help us along the way. Hölder’s inequality. Most plainly, for sums this is equivalent to:

In other words, if p,q are positive reals and aᵢ, bᵢ are non-negative reals then have the following inequality.

However, this still applies to cyclic sums. Let us rewrite our inequality again. We know that:

And similarly we can write:

Why would we do this? Notice that this is very similar to the denominator in our original cyclic sum. Matter of fact, we can now rewrite this inequality as:

Which is what we will now show. Let us use Hölder on just the first two terms first. Then we have that:

Using it once more we have that:

But now the right hand side just evaluates to (1+1+1)³ = 27. Therefore we have:

And putting everything together, we have:

Or in our original form,

As required.

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