2024 AMC 10B Problem 9

Attempt Problem 9 of the 2024 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 10B solutions, or check the answer key.

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Real numbers $a, b,$ and $c$ have arithmetic mean $0.$ The arithmetic mean of $a^2, b^2,$ and $c^2$ is $10.$ What is the arithmetic mean of $ab, ac,$ and $bc?$

$-5$

$-\dfrac{10}{3}$

$-\dfrac{10}{9}$

$0$

$\dfrac{10}{9}$

Answer: A

Concepts:algebraic manipulation mean

Difficulty rating: 1350

Solution:

The means tell us $a + b + c = 0$ and $a^2 + b^2 + c^2 = 30.$ Square the first: $(a+b+c)^2$ $= a^2 + b^2 + c^2$ $+ 2(ab + bc + ca),$ so $0 = 30 + 2(ab + bc + ca)$ and $ab + bc + ca = -15.$ Their mean is $-15 / 3 = -5.$ Thus, A is the correct answer.

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