Let $f(x) = \lfloor x \lfloor x \rfloor \rfloor$ for $x \ge 0.$ (a) Find all $x \ge 0$ such that $f(x) = 1.$

(b) Find all $x \ge 0$ such that $f(x) = 3.$

(c) Find all $x \ge 0$ such that $f(x) = 5.$

(d) Find the number of possible values of $f(x)$ for

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