ABC Triples

A,B, and C are real numbers. How many triples (A,B,C) exist such that AB=C, AC=B, and BC=A?
Source: mathcontest.olemiss.edu 3/3/2014

SOLUTION
AC=B
Divide both sides by A assuming A\neq 0
C=\dfrac{B}{A}

BC=A
Divide both sides by B assuming B\neq 0
C=\dfrac{A}{B}

\dfrac{B}{A}=\dfrac{A}{B}
ConsiderĀ x=\dfrac{B}{A} a real number
x=\dfrac{1}{x}
x^2=1
x=\pm 1
The triples (A,B,C) are
1) (0,0,0)
2) (1,1,1)
3) (1,-1,-1)
4) (-1,1,-1)
5) (-1,-1,1)

Answer: 5

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About mvtrinh

Retired high school math teacher.
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