<s>[INST] A $4$ variable inequality If $a,b,c,d$ are positive numbers such that $c^2+d^2=(a^2+b^2)^3$, prove that 
$$\frac{a^3}{c} + \frac{b^3}{d} \ge 1,$$
with equality if and only if $ad=bc$.
Source: Don Sokolowsky, Crux Mathematicorum, Vol. 6, No. 8, October 1980, p.259.
 [/INST] Because by Holder
$$\left(\frac{a^3}{c}+\frac{b^3}{d}\right)^2(c^2+d^2)\geq(a^2+b^2)^3$$
and we are done!
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