## 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 $B\neq 0$
$C=\dfrac{A}{B}$

$\dfrac{B}{A}=\dfrac{A}{B}$
Consider $x=\dfrac{B}{A}$ 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: $5$