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.
by
Finn Hulse