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XV-th Junior Balkan Mathematical Olympiad

Algebra Difficulty 4.6 AIME Prove it North Macedonia

Let $a$ , $b$ and $c$ be positive real numbers such that $abc = 1$ . Prove the inequality
$(a^5 + a^4 + a^3 + a^2 + a + 1)(b^5 + b^4 + b^3 + b^2 + b + 1)(c^5 + c^4 + c^3 + c^2 + c + 1) \geq 8(a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1)$

Solution

By factorizing we get
$(a^5 + a^4 + a^3 + a^2 + a + 1) = (a^3 + 1)(a^2 + a + 1).$
We apply same thing to the other terms and simply to get $(a^3 + 1)(b^3 + 1)(c^3 + 1) \geq 8$ .
By $AM \geq GM$ we have
$a^3 + 1 \geq 2\sqrt{a^3}$
$b^3 + 1 \geq 2\sqrt{b^3}$
$c^3 + 1 \geq 2\sqrt{c^3}$
$(a^3 + 1)(b^3 + 1)(c^3 + 1) \geq 8\sqrt{a^3 b^3 c^3}$
Equality holds if and only if $a = b = c = 1$ .

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