Maximize a Constant in an Inequality with a Restriction

Algebra Level 4

Real numbers \(a\), \(b\), and \(c\) satisfy \(abc=1\). Given that \(k\) is an integer \(\geq 2\), find the maximum possible \(R\) that satisfies

\[\dfrac{a^k}{a+b}+\dfrac{b^k}{b+c}+\dfrac{c^k}{c+a} \geq R\]

for all \(a\), \(b\), and \(c\) satisfying the condition previously stated.

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