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Find f(x) for every rational x

Source: IMO LongList 1979 - P27

September 15, 2010
algebrabinary representationfunctional equationfunctionIMO ShortlistIMO Longlist

Problem Statement

For all rational xx satisfying 0≤x<10 \leq x < 1, the functions ff is defined by f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases} Given that x=0.b1b2b3⋯x = 0.b_1b_2b_3 \cdots is the binary representation of xx, find, with proof, f(x)f(x).
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