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, 2023

Algebra Difficulty 7.8 National olympiad, round 2 Prove it Turkey

Find the smallest value of
$xy^3z^2 + \frac{4z}{x} - 8yz - \frac{4}{xy}$
where $x$ , $y$ , $z$ are positive real numbers satisfying at least one of the following inequalities:
$2xy > 1 \text{ and } yz > 1.$

Solution

4. For any $x > 0$ , we have
$P(x) = x^n Q(1/x) \ge Q(x) = P(1/x) x^n \ge x^n Q(1/x) = P(x)$
hence we have equalities in each step so $P(x) = Q(x)$ for all positive values of $x$ , hence for all $x$ .

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