Question: For all real numbers $ a, b, c $, find the number of functions $ f : \mathbb{R} \to \mathbb{R} $ such that
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- Equality holds when $ \frac{x}{y} = \frac{y}{z} = \frac{z}{x} $, which implies $ x = y = z $. Since $ x + y + z = 1 $, we get $ x = y = z = \frac{1}{3} $.
- Check the expression at $ x = y = z = \frac{1}{3} $:
- \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x} = 3 \cdot \frac{(1/3)^2}{1/3} = 3 \cdot \frac{1/9}{1/3} = 3 \cdot \frac{1}{3} = 1
- f(a + b + c) = f(a) + f(b) + f(c)
- and
- f(ab) = f(a)f(b)
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