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}.

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