# Playing Carts - II

Algebra Level pending

Consider a Cartesian plane where the functions $$f(x) = x^3, \; f'(x), \; f''(x)$$ are all plotted.

Let $$\triangle{\alpha}$$ be the triangle with vertexes on all lattice points for which one of the functions intersects another.

Let $$\triangle{\beta}$$ be the triangle with vertexes on the first quadrant points for which one of the functions intersects another.

(1) The distance between $$\alpha$$'s midpoint to the origin of the Cartesian plane is greater than the one between $$\beta$$'s midpoint to the origin.

(2) $$\alpha$$ contains the greatest side length compared to $$\beta$$.

(4) Both triangles are obtuse.

(8) $$\alpha$$'s circumradius is greater than $$\beta$$'s.

(16) The parabola passing through all $$\beta$$'s vertexes has real roots.

Find the sum of the numbers in all correct given statements

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