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Positive real $a$, $b$, $c$, and $d$ are such that $a + b + c + d = 1$, find the minimum of:

$\frac{a^3}{b + c} + \frac{b^3}{c + d} + \frac{c^3}{d + a} + \frac{d^3}{a + b} .$

This is part of the set My Problems and THRILLER

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