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2025 AMC 10A Problem 18

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2025 AMC 10A Problem 18AMC 10 Step by Step
Name:                             Date: September 13, 2026
Quadratics · difficulty 3 of 5 · about 4 min · position P16-20

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

113(14+14+15)=307\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

∏k=12025(kx2−4x−3)=(x2−4x−3)(2x2−4x−3)(3x2−4x−3)…(2025x2−4x−3)?\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) ?

  1. A)

    −53-\frac{5}{3}

  2. B)

    −32-\frac{3}{2}

  3. C)

    −65-\frac{6}{5}

  4. D)

    −56-\frac{5}{6}

  5. E)

    −23-\frac{2}{3}

Working