a^4 + 1b^3 + b^2 + b + b^4 + 1c^3 + c^2 + c + c^4 + 1a^3 + a^2 + a 2.

a^4+1b^3+b^2+b + b^4+1c^3+c^2+c + c^4+1a^3+a^2+a 3 [3]827 = 2.

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Team Selection Test for JBMO 2023

Turkey 2023 algebra

Problem

\frac{a ^{4} + 1}{b ^{3} + b ^{2} + b} + \frac{b ^{4} + 1}{c ^{3} + c ^{2} + c} + \frac{c ^{4} + 1}{a ^{3} + a ^{2} + a} \geq 2.

Solution

\frac{a ^{4} + 1}{b ^{3} + b ^{2} + b} + \frac{b ^{4} + 1}{c ^{3} + c ^{2} + c} + \frac{c ^{4} + 1}{a ^{3} + a ^{2} + a} \geq 3 \cdot 3 \frac{8}{27} = 2.

Techniques

QM-AM-GM-HM / Power Mean