INMO 2016 proposal from IISc

Algebra Level 5

f(x)2(1+x)x[1,)xf(x+1)=(f(x))21x[1,)\begin{aligned} f(x) & \le 2(1+x) & \forall x \in [1, \infty) \\ x f (x+1) & = (f(x))^2 - 1 & \forall x \in [ 1,\infty ) \end{aligned}

A function f:[1,)[1,)f:[1, \infty) \rightarrow [1, \infty) satisfy the conditions above. Find f(2016)f(2016).

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