Difference between revisions of "2024 AMC 10B Problems/Problem 9"

Latest revision as of 01:21, 20 November 2024

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 $30 + 2ab + 2ac + 2bc = 0$ Arithmetic will give you that $ac + bc + ac = -15$ . To find the arithmetic mean, divide that by 3, so $\frac{ac + bc + ac}{3} = \boxed{\textbf{(A) }-5}$

~ARay10 [Feel free to clean this up!]

~Mr.Lightning [Cleaned it up a bit]

Solution 2

Since $\frac{a+b+c}{3},$ we have $a+b+c=0,$ and $(a+b+c)^2= a^2 + b^2+c^2+2(ab+ac+bc)=0$

From the second given, $\frac{a^2+b^2+c^2}{3} = 10$ , so $a^2+b^2+c^3=30.$ Substituting this into the above equation, $2(ab+ac+bc) = (a+b+c)^2 -(a^2+b^2+c^2)=0-30 = -30.$ Thus, $ab+ac+bc=-15,$ and their arithmetic mean is $\frac{-15}{3} = \boxed{\textbf{(A)}\ -5}.$

~laythe_enjoyer211, countmath1

Solution 3

Assume that $a = 0$ and $b = -c$ . Since we know that the arithmetic mean of the three numbers is $10$ , this means $a^2+b^2+c^2=30$ . Using this equation, $b^{2} + b^{2} = 30$ , so $b^2 = 15$ . Observe that taking the positive or negative root won't matter as $c$ will be the opposite. If we let $b = \sqrt{15}$ and $c = -\sqrt{15}$ , $ab = 0\times\sqrt{15} = 0$ , $ac = 0\times-\sqrt{15} = 0$ , and $bc = \sqrt{15}\times-\sqrt{15} = -15$ , so $ab+ac+bc=-15$ . Doing $\frac{-15}{3}$ to get the arithmetic mean will give us $\boxed{\textbf{(A)}\ -5}$ .

-aleyang

~unpogged (cleaned it up, fixed some errors)

🎥✨ Video Solution by Scholars Foundation ➡️ (Easy-to-Understand 💡✔️)

https://youtu.be/T_QESWAKUUk?si=yFqIs-XfQ878b9bg&t=520

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

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

~ Pi Academy

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=24EZaeAThuE

@@ Line 6: / Line 6: @@
 ==Solution 1==
-If <math>\frac{a+b+c}{3} = 0</math>, that means <math>a+b+c=0</math>, and <math>(a+b+c)^2=0</math>. Expanding that gives <math>(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc</math>. If <math>\frac{a^2+b^2+c^2}{3} = 10</math>, then <math>a^2+b^2+c^2=30</math>. Thus, we have <math>0 = 30 + 2ab + 2ac + 2bc</math>. Arithmetic will give you that <math>ac + bc + ac = -15</math>. To find the arithmetic mean, divide that by 3, so <math>\frac{ac + bc + ac}{3} = \boxed{\textbf{(A) }-5}</math>
+If <math>\frac{a+b+c}{3} = 0</math>, that means <math>a+b+c=0</math>, and <math>(a+b+c)^2=0</math>. Expanding that gives <cmath>(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc</cmath> If <math>\frac{a^2+b^2+c^2}{3} = 10</math>, then <math>a^2+b^2+c^2=30</math>. Thus, we have <cmath>30 + 2ab + 2ac + 2bc = 0</cmath> Arithmetic will give you that <math>ac + bc + ac = -15</math>. To find the arithmetic mean, divide that by 3, so <math>\frac{ac + bc + ac}{3} = \boxed{\textbf{(A) }-5}</math>
+~ARay10 [Feel free to clean this up!]
+~Mr.Lightning [Cleaned it up a bit]
 ==Solution 2==
@@ Line 20: / Line 24: @@
 ==Solution 3==
-Assume that <math>a = 0</math> and <math>b = -c</math>. Plugging into the second equation, <math>b^{2} + b^{2} = 30</math>, so <math>b^2 = 15</math>. Observe that taking the positive or negative root won't matter, as c will be the opposite. If we let <math>b = \sqrt{15} and </math>c = -\sqrt{15}, <math>0\times\sqrt{15} + 0\times-\sqrt{15} + \sqrt{15}-\sqrt{15} is -15, and dividing by 3 gives us </math>\boxed{\textbf{(A)}\ -5}.$
+Assume that <math>a = 0</math> and <math>b = -c</math>. Since we know that the arithmetic mean of the three numbers is <math>10</math>, this means <math>a^2+b^2+c^2=30</math>. Using this equation, <math>b^{2} + b^{2} = 30</math>, so <math>b^2 = 15</math>. Observe that taking the positive or negative root won't matter as <math>c</math> will be the opposite. If we let <math>b = \sqrt{15}</math> and <math>c = -\sqrt{15}</math>, <math>ab = 0\times\sqrt{15} = 0</math>, <math>ac = 0\times-\sqrt{15} = 0</math>, and <math>bc = \sqrt{15}\times-\sqrt{15} = -15</math>, so <math>ab+ac+bc=-15</math>. Doing <math>\frac{-15}{3}</math> to get the arithmetic mean will give us <math>\boxed{\textbf{(A)}\ -5}</math>.
+-aleyang
+~unpogged (cleaned it up, fixed some errors)
+==🎥✨ Video Solution by Scholars Foundation ➡️ (Easy-to-Understand 💡✔️)==
+https://youtu.be/T_QESWAKUUk?si=yFqIs-XfQ878b9bg&t=520
 ==Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)==
@@ Line 27: / Line 39: @@
 ~ Pi Academy
+==Video Solution 2 by SpreadTheMathLove==
+https://www.youtube.com/watch?v=24EZaeAThuE
 ==See also==
 {{AMC10 box|year=2024|ab=B|num-b=8|num-a=10}}
 {{MAA Notice}}

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