AMC 10A/12A 2025 (Problem 18/12)

The harmonic mean of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers in the collection. For example, the harmonic mean of

4,4,

and

5

\frac{1}{\frac13\left(\frac14+\frac14+\frac15\right)} = \frac{30}{7}

What is the harmonic mean of all the real roots of the 4050th-degree polynomial

\prod_{k=1}^{2025} \left(kx^2-4x-3\right) = (x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)\cdots(2025x^2-4x-3)?

\text{(A)}\;-\frac{5}{3}

\text{(B)}\;-\frac{3}{2}

\text{(C)}\;-\frac{6}{5}

\text{(D)}\;-\frac{5}{6}

\text{(E)}\;-\frac{2}{3}