Question

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

Small Hint:

Mean $0$ means $a + b + c = 0;$ mean $10$ means $a^2 + b^2 + c^2 = 30$

Big Hint:

Use $(a + b + c)^2$ $= a^2 + b^2 + c^2$ $+ 2(ab + bc + ca)$

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 $-\frac{15}{3} = -5.$ Thus, A is the correct answer.

2024 AMC 10B Problem 9

Answer: A

Difficulty rating: 1350

Small Hint:

Big Hint:

Solution:

Problem 9 in Other Years