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