Maximize a Constant in an Inequality with a Restriction

Algebra Level 4

Real numbers aa, bb, and cc satisfy abc=1abc=1. Given that kk is an integer 2\geq 2, find the maximum possible RR that satisfies

aka+b+bkb+c+ckc+aR\dfrac{a^k}{a+b}+\dfrac{b^k}{b+c}+\dfrac{c^k}{c+a} \geq R

for all aa, bb, and cc satisfying the condition previously stated.

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