AMC 10B 2024 (Problem 9)Real numbers aaa, bbb, and ccc have arithmetic mean 000. The arithmetic mean of a2a^2a2, b2b^2b2, and c2c^2c2 is 101010. What is the arithmetic mean of ababab, acacac, and bcbcbc?(A) −5\text{(A)}\;-5(A)−5(B) −103\text{(B)}\;-\frac{10}{3}(B)−310(C) −109\text{(C)}\;-\frac{10}{9}(C)−910(D) 0\text{(D)}\;0(D)0(E) 109\text{(E)}\;\frac{10}{9}(E)910Related TopicsCoreAlgebraic IdentitiesMinorMean, Median, Mode, and RangeCheck AnswerYour answer:ABCDECheckHints (3)Hint 1a+b+c3=0⇒a+b+c=0\frac{a+b+c}{3}=0\quad\Rightarrow\quad a+b+c=03a+b+c=0⇒a+b+c=0a2+b2+c23=10⇒a2+b2+c2=30\frac{a^2+b^2+c^2}{3}=10\quad\Rightarrow\quad a^2+b^2+c^2=303a2+b2+c2=10⇒a2+b2+c2=30Hint 2(a+b+c)2=a2+b2+c2+2ab+2ac+2bc(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc(a+b+c)2=a2+b2+c2+2ab+2ac+2bcHint 30=30+2(ab+ac+bc)0=30+2(ab+ac+bc)0=30+2(ab+ac+bc)ab+ac+bc=−15ab+ac+bc=-15ab+ac+bc=−15ab+ac+bc3=−5\frac{ab+ac+bc}{3}=-53ab+ac+bc=−5Final Answer(A) −5-5−5Related Problems (4)AMC 10B Fall 2021 (Problem 22)USAJMO 2023 (Problem 1)USAMO 2018 (Problem 1)AMC 12A 2024 (Problem 15)