Team Selection Test for EGMO 2023

Question

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

Accepted Answer

4. For any x > 0, we have P(x) = x^n Q(1/x) Q(x) = P(1/x) x^n 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.