Fractions Fun with Functions

Algebra Level 5

Let \(f(x)\) be a polynomial with integer coefficients defined by \(f:\mathbb{R}^+\to \mathbb{R}^+\) satisfying \[\dfrac{f(x)}{x}=\sqrt{\dfrac{f(f(x))}{f(x)}}\]

Find the sum of all possible values of \(f(3)\) subject to \(f(3) < 100\).

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