2024 AMC 10B Problems/Problem 9
Contents
[hide]Problem
Real numbers and have arithmetic mean 0. The arithmetic mean of and is 10. What is the arithmetic mean of and ?
Solution 1
If , that means , and . Expanding that gives If , then . Thus, we have Arithmetic will give you that . To find the arithmetic mean, divide that by 3, so
~ARay10 [Feel free to clean this up!]
~Mr.Lightning [Cleaned it up a bit]
Solution 2
Since we have and
From the second given, , so Substituting this into the above equation, Thus, and their arithmetic mean is
~laythe_enjoyer211, countmath1
Solution 3
Assume that and . Since we know that the arithmetic mean of the three numbers is , this means . Using this equation, , so . Observe that taking the positive or negative root won't matter as will be the opposite. If we let and , , , and , so . Doing to get the arithmetic mean will give us .
-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
See also
2024 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.