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?

2024 AMC 10B Problem 9

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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

10.

What is the arithmetic mean of

ab,

ac,

and

bc?

-5

-\dfrac{10}{3}

-\dfrac{10}{9}

0

\dfrac{10}{9}

Mean

0

means

a + b + c = 0;

mean

10

means

a^2 + b^2 + c^2 = 30

Use

(a + b + c)^2

= a^2 + b^2 + c^2

+ 2(ab + bc + ca)

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.