## 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 $kabc > a^{3}+b^{3}+c^{3}$, 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, $c > 0$ with $a + b \leq c$, we have $kabc \leq a^{3}+b^{3}+c^{3}$. First take $b=a$ and $c=2a$. Then, $2ka^{3} \leq 10a^{3}$, that is, $k \leq 5$. Conversely, let $k=5$. Set $c=a+b+x$, where $x \geq 0$. Then,

$a^{3}+b^{3}+c^{3}-5abc =2(a+b)(a-b)^{2}+(ab+3a^{2}+3b^{2})x+3(a+b)x^{2}+x^{3} \geq 0$.

QED.

More later,

Nalin Pithwa

This site uses Akismet to reduce spam. Learn how your comment data is processed.