2023 AMC 12A Problems/Problem 23

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Problem

How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation \[(1+2a)(2+2b)(2a+b) = 32ab?\]

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{an infinite number}$

Solution 1: AM-GM Inequality

Using AM-GM on the two terms in each factor on the left, we get \[(1+2a)(2+2b)(2a+b) \ge 8\sqrt{2a \cdot 4b \cdot 2ab}= 32ab,\] meaning the equality condition must be satisfied. This means $1 = 2a = b$, so we only have $\boxed{1}$ solution.

Solution 2: Sum Of Squares

Equation $(1+2a)(2+2b)(2a+b)=32ab$ is equivalent to \[b(2a-1)^2+2a(b-1)^2+(2a-b)^2=0,\] where $a$, $b>0$. Therefore $2a-1=b-1=2a-b=0$, so $(a,b)=\left(\tfrac12,1\right)$. Hence the answer is $\boxed{\textbf{(B) }1}$.

Video Solution 1 by OmegaLearn

https://youtu.be/LP4HSoaOCSU

Video Solution by MOP 2024

https://youtu.be/kkx7sm6-ZE8

~r00tsOfUnity

Video Solution

https://youtu.be/ZKdnv8MsEDI

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

See also

2023 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
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 12 Problems and Solutions

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