2024 AMC 10B Problem 9

Problem 9 of 25EasierAlgebra

Real numbers

a,

b,

and

c

have arithmetic mean

0.

The arithmetic mean of

a^2,

b^2,

and

c^2

10.

What is the arithmetic mean of

ab,

ac,

and

bc?

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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),

0 = 30 + 2(ab + bc + ca)

and

ab + bc + ca = -15.

Their mean is

-\frac{15}{3} = -5.

Thus, A is the correct answer.