AMC 10 · 2025 · #18

Grade 7 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
vieta-formulasquadratic-equationsfraction-arithmetic easier-related-problem ↑ Prerequisites: vieta-formulasquadratic-equations
📏 Medium solution 💡 2 insights
Problem
The harmonic mean of a list of numbers is the reciprocal of the average of their reciprocals. A giant polynomial of degree 4050 is written as a product of 2025 quadratics; the k-th factor is k*x² - 4x - 3, for k running from 1 to 2025. Find the harmonic mean of all of its real roots.

Pick an answer.

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

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Solve an Easier Related Problem

Finding all 4050 roots is hopeless, so change what we chase: the harmonic mean depends only on the sum of the reciprocals of the roots, not the roots themselves. That sum splits factor by factor, so we solve one easy quadratic instead of the monster. For a single quadratic, the sum of the reciprocals of its two roots equals -b/c, which here erases k entirely. Every one of the 2025 factors then contributes the same amount, and a short multiplication and division finish the job.

1STEP 1

Rewrite the harmonic mean

Flipping the average of reciprocals turns the harmonic mean into n divided by their sum. Here n = 4050, so all we need is the sum of 1/root.

HM=1/1/nΣ_i=1ⁿ1/x_i=n/Σ_i=1ⁿ1/x_i, n=4050
2STEP 2

Shrink it to one quadratic

A product just pools its factors' roots, so the reciprocal sum splits into 2025 copies of one quadratic's reciprocal sum.

Σ_i=1⁴⁰⁵⁰1/x_i=Σ_k=1²⁰²⁵(1/r_k+1/s_k), r_k,s_k are the roots of kx²-4x-3
3STEP 3

Reciprocal sum for one quadratic

Vieta on kx² - 4x - 3 gives r + s = 4/k and rs = -3/k, so 1/r + 1/s = (r+s)/(r*s) = -4/3 and the k cancels.

1/r+1/s=(r+s)/rs=(4/k)/(-3/k)=-4/3
4STEP 4

Same answer for every k

The reciprocal sum is -b/c, and b = -4, c = -3 in every factor, so all 2025 give -4/3: the total is -2700.

Σ_k=1²⁰²⁵(-4/3)=2025×(-4/3)=-2700
5STEP 5

Divide to get the harmonic mean

Back into the Step 1 formula: 4050 divided by -2700, and both share 1350, so the harmonic mean is -3/2 — choice (B).

HM=4050/-2700=-4050/2700=-3/2
Answer
-3/2
Test the smallest case by hand. The k = 1 factor is x² - 4x - 3, whose roots are 2 + sqrt(7) and 2 - sqrt(7); their reciprocal-sum is (2+sqrt7 + 2-sqrt7)/((2)² - 7) = 4/-3 = -4/3, and their harmonic mean is 2/-4/3 = -3/2. Since every quadratic contributes the identical -4/3, stacking all 2025 of them keeps the harmonic mean at -3/2. The value is negative, which fits the fact that -3 in each factor forces a negative product of roots, so it is believable and equals choice (B).
💡Key takeaway

When a mean only cares about the reciprocals of the roots, you can skip solving and use that the reciprocal-sum of a quadratic's roots is just -b/c.

  • Rewrite the harmonic mean
  • Shrink it to one quadratic
  • Reciprocal sum for one quadratic
  • Same answer for every k
  • Divide to get the harmonic mean

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