2025 AMC 10A Problem 18

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

\[\frac{1}{\frac{1}{3}\left(\frac{1}{4} + \frac{1}{4} + \frac{1}{5}\right)} = \frac{30}{7}.\]

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

\[ \begin{gathered} \prod_{k=1}^{2025}(kx^2 - 4x - 3) \\ {}= (x^2 - 4x - 3) \\ \quad {}\cdot (2x^2 - 4x - 3) \\ \quad {}\cdot (3x^2 - 4x - 3)\cdots \\ \quad {}\cdot (2025x^2 - 4x - 3)? \end{gathered} \]

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

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.