## Weird Inequality

Prove that given we have

.

**Solution:**

It’s easy to see that the inequality is equivalent to . ( the original form is just for the impression 🙂 )

It may seem difficult to work with these nasty radicals, but there is a trick that blows up the whole thing… 🙂

Let’s see what happens if we replace with where ? We get both sides multiplied with , which shows that if we prove the inequality for one of the triplets then the inequality is true for also. Picking , we get , so if we prove the inequality for we are done.

Therefore, we have to prove that , for all . This is an easy calculus exercise, because, fixing we get a function which has a strictly decreasing derivative, so it has a unique minimum point, which is positive.

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Categories: Inequalities, Problem Solving
inequality, trick

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