# Using the Problem to Solve the Problem

Question:

Find $(a/c)^3$ if $\frac{1}{a+c}=\frac{1}{a}+\frac{1}{c}$. (Source: Mandelbrot)

Official Solution:

Multiplying both sides by $ac(a+c)$, we get $ac=a(a+c)+c(a+c)=a^2+2ac+c^2$, so $a^2+ac+c^2=0$. Multiplying by $a-c$ gives $(a-c)(a^2+ac+c^2)=0$, so $a^3-c^3=0$, and thus $(a/c)^3=a^3/c^3=\boxed{1}$.

I think we can do much better than that:

$a$ and $c$ have symmetric roles in the equation. Thus, if $(a,c)$ is a solution, so is $(c,a)$. Since the questions asks to find $(a/c)^3$, there must be only one value it equals regardless of $a$ or $c$. Thus, the sought amount equals its reciprocal, and is thus either 1 or -1. Since the latter causes a zero denominator in the left hand side, the answer must be the only choice left, $\boxed{1}$.