2025 AMC 12A Problem 12

Problem 12 of 25IntermediateAlgebra

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}{\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

4050

th degree polynomial

\begin{aligned} &\small \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 (2025x^2 - 4x - 3)? \end{aligned}

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Solution

Each factor

kx^2 - 4x - 3

has discriminant

16 + 12k \gt 0,

so it has two real roots; there are

4050

roots in all. For the roots of

kx^2 - 4x - 3,

the sum of reciprocals is

\dfrac{\text{sum}}{\text{product}} = \dfrac{\frac{4}{k}}{-\frac{3}{k}} = -\dfrac{4}{3},

independent of

k.

Summing over all

2025

factors,

\displaystyle\sum \frac{1}{r} = 2025\left(-\frac{4}{3}\right) = -2700.

The harmonic mean is

\frac{4050}{-2700} = -\frac{3}{2}.

Thus, the correct answer is B.

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