Problem

Source: China TST 2009, Quiz 2, Problem 3

Tags: function, algebra proposed, algebra



Consider function $ f: R\to R$ which satisfies the conditions for any mutually distinct real numbers $ a,b,c,d$ satisfying $ \frac {a - b}{b - c} + \frac {a - d}{d - c} = 0$, $ f(a),f(b),f(c),f(d)$ are mutully different and $ \frac {f(a) - f(b)}{f(b) - f(c)} + \frac {f(a) - f(d)}{f(d) - f(c)} = 0.$ Prove that function $ f$ is linear