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2024 AMC 10B Problem 9

Below is the professionally curated solution for Problem 9 of the 2024 AMC 10B, 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.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:algebraic manipulationmean

Difficulty rating: 1350

9.

Real numbers a,b,a, b, and cc have arithmetic mean 0.0. The arithmetic mean of a2,b2,a^2, b^2, and c2c^2 is 10.10. What is the arithmetic mean of ab,ac,ab, ac, and bc?bc?

−5-5

−103-\dfrac{10}{3}

−109-\dfrac{10}{9}

00

109\dfrac{10}{9}

Solution:

The means tell us a+b+c=0a + b + c = 0 and a2+b2+c2=30.a^2 + b^2 + c^2 = 30. Square the first: (a+b+c)2=a2+b2+c2+2(ab+bc+ca),(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca), so 0=30+2(ab+bc+ca)0 = 30 + 2(ab + bc + ca) and ab+bc+ca=−15.ab + bc + ca = -15. Their mean is −15/3=−5.-15 / 3 = -5. Thus, A is the correct answer.

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