# When is this an Integer?

Algebra Level 5

$\sqrt[3] {\text{min}\left[\prod_{cyc}\left(\dfrac{ax^2+bx}{c}\right)\right]}$

Let $$0< x\le 1000$$ be a real constant. Find the number of distinct $$x$$'s such that, for positive real variables $$a,b,c$$ the expression above is an integer.

Note: $$a,b,c$$ are variables, $$x$$ is a constant. The minimum value is taken where $$a,b,c$$ vary over positive reals.

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