## Triangle inequality and max-min

Here is a problem I culled from Prof. Titu Andreescu’s literature (Geometric maxima and minima) for IITJEE Mathematics:

**Problem:**

Find the greatest real number k such that for any triple of positive numbers a, b, c such that , there exists a triangle with side lengths a, b, c.

**Solution:**

We have to find the greatest real number k such that for any a, b, with , we have . First take and . Then, , that is, . Conversely, let . Set , where . Then,

.

**QED.**

More later,

Nalin Pithwa

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