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Problem 1747

National Olympiad, first round

Algebra Difficulty 6.1 Prove it Fall Mathematical Competition · Bulgaria

Find all real numbers $r$ such that the inequality
$r(ab + bc + ca) + (3 - r) \left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \ge 9$
holds true for arbitrary positive numbers $a, b$ and $c$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Taking $a = b = c$ we obtain
$ra^2 + (3 - r) \frac{1}{a} \ge 3 \iff (a - 1)(r(a^2 + a + 1) - 3) \ge 0$
for any $a > 0$ . Then it easily follows that $r = 1$ .

Conversely, let $r = 1$ . Then we write the inequality as $\frac{2 + abc}{3} \ge \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}$ . The GM-HM inequality implies that the right hand side does not exceed $\sqrt[3]{abc}$ and, setting $x = \sqrt[3]{abc} > 0$ , it is enough to prove that
$\frac{2+x^3}{3} \ge x$
which is equivalent to the obvious $(x-1)^2(x+2) \ge 0$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.