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x + y^2 + z^3 + w^6 >=a (xyzw)^b

Source: Switzerland - 2008 Swiss MO Final Round p10

December 26, 2022

algebrainequalities

Problem Statement

Find all pairs

(a, b)

of positive real numbers with the following properties: (i) For all positive real numbers

x, y, z,w

holds

x + y^2 + z^3 + w^6 \ge a (xyzw)^{b}

. (ii) There is a quadruple

(x, y, z,w)

of positive real numbers such that in equality (i) applies.