AMC 12 2025a Problem 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 $5$ is

$$\frac{1}{\frac{1}{3}(\frac{1}{4}+\frac{1}{4}+\frac{1}{5})}=\frac{30}{7}.$$

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

$$\prod_{k=1}^{2025} (kx^2-4x-3)=(x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)...(2025x^2-4x-3)?$$

\textbf{(A)}-\frac{5}{3} \textbf{(B)}-\frac{3}{2} \textbf{(C)}-\frac{6}{5} \textbf{(D)}-\frac{5}{6} \textbf{(E)}~-\frac{2}{3}