2024 AMC 10B Problems/Problem 9

Problem

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 $bc$ ?

$\textbf{(A) } -5 \qquad\textbf{(B) } -\dfrac{10}{3} \qquad\textbf{(C) } -\dfrac{10}{9} \qquad\textbf{(D) } 0 \qquad\textbf{(E) } \dfrac{10}{9}$

Solution 1

If $\frac{a+b+c}{3} = 0$ , that means $a+b+c=0$ , and $(a+b+c)^2=0$ . Expanding that gives $(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc$ . If $\frac{a^2+b^2+c^2}{3} = 10$ , then $a^2+b^2+c^2=30$ . Thus, we have $0 = 30 + 2ab + 2ac + 2bc$ . Arithmetic will give you that $ac + bc + ac = -15$ . To find the arithmetic mean, divide that by 3, so $\frac{ac + bc + ac}{3} = \boxed{\textbf{(A) }-5}$

Solution 2

Given: $\frac{a+b+c}{3}=0$

$\Rightarrow a+b+c=0$

Square both sides to get: $(a+b+c)^2=0$

$a^2+b^2+c^2+2(ab+bc+ca)=0 \longrightarrow \raisebox{.5pt}{\textcircled{\raisebox{-.9pt}{1}}}$

Also given: $\frac{a^2+b^2+c^2}{3} = 10$

$\Rightarrow a^2+b^2+c^2 = 30$

Substituting into equation $\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}}$ ,

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

$2(ab+bc+ca)=-30$

$ab+bc+ca=-15$

There are 3 terms, so the mean is: $\frac{ab+bc+ca}{3}$

$= \frac{-15}{3} = \boxed{\textbf{(A) }-5}$

~laythe_enjoyer211

Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)

https://youtu.be/QLziG_2e7CY?feature=shared

~ Pi Academy

2024 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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2024 AMC 10B Problems/Problem 9

Contents

Problem

Solution 1

Solution 2

Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)

See also