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2024 AMC 10B Problem 9

Problem 9 of 25EasierAlgebraProbability & Statistics

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

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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(a+b+c)^2 =a2+b2+c2= a^2 + b^2 + c^2 +2(ab+bc+ca),+ 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 −153=−5.-\frac{15}{3} = -5. Thus, A is the correct answer.
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