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

 

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