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F(x/3) = F(x)/2, F(1-x)=1-F(x), F( 173/1993) ?, F(1/13)?

Source: Nordic Mathematical Contest 1993 #1

October 5, 2017
functional equationincreasing functionsalgebra

Problem Statement

Let FF be an increasing real function defined for all x,0≤x≤1x, 0 \le x \le 1, satisfying the conditions (i) F(x3)=F(x)2F (\frac{x}{3}) = \frac{F(x)}{2}. (ii) F(1−x)=1−F(x)F(1- x) = 1 - F(x). Determine F(1731993)F(\frac{173}{1993}) and F(113)F(\frac{1}{13}) .
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