The $\textit{harmonic\ mean}$ 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 4050th degree polynomial
$$\prod_{k=1}^{2025} (kx^2-4x-3) = (x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)\dots (2025x^2-4x-3) ?$$
$\textbf{(A) } -\frac{5}{3} \qquad\textbf{(B) } -\frac{3}{2} \qquad\textbf{(C) } -\frac{6}{5} \qquad\textbf{(D) } -\frac{5}{6} \qquad\textbf{(E) } -\frac{2}{3}$
Difficulty: 61/100 — Although easy for experienced competitors, it applies Vieta's formulas in a nontraditional way, making it hard to approach for beginners.
Core Concepts: Vieta's formulas
Challenges: Using what you know about the roots to determine a value for the sum of reciprocals. Figuring out a way to apply Vieta's formulas to the harmonic mean.
Techniques: Treat each factor separately, computing the desired sum in parts.
Error-prone Steps: Attempting to expand the polynomial and directly use Vieta's formulas on the entire polynomial.
Ideal Time:
Experienced: ≤ 30-60 sec
Intermediate: ≤ 1.5-2 min
Beginner: ≤ 4-5 min