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Determine f(1/7)

Source: IMO Longlist 1989, Problem 17

September 18, 2008
functionlimitgeometric seriesalgebra unsolvedalgebra

Problem Statement

Let a∈R,0<a<1, a \in \mathbb{R}, 0 < a < 1, and f f a continuous function on [0,1] [0, 1] satisfying f(0) \equal{} 0, f(1) \equal{} 1, and f \left( \frac{x\plus{}y}{2} \right) \equal{} (1\minus{}a) f(x) \plus{} a f(y)   \forall x,y \in [0,1] \text{ with } x \leq y. Determine f(17). f \left( \frac{1}{7} \right).
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