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periodic if f (x) + f (x + 2a) + f (x + 3a) + f (x + 5a) = 1

Source: 12th or 13th QEDMO problem 4 (11. - 15. 12. 2013) https://artofproblemsolving.com/community/c2400093_2013_qedmo_13th_or_12th

July 5, 2021
periodicfunctionalfunctional equationalgebra

Problem Statement

Let a>0a> 0 and f:R→Rf: R\to R a function such that f(x)+f(x+2a)+f(x+3a)+f(x+5a)=1f (x) + f (x + 2a) + f (x + 3a) + f (x + 5a) = 1 for all x∈Rx\in R . Show that ff is periodic, that is, that there is some b>0b> 0 for which f(x)=f(x+b)f (x) = f (x + b) for every x∈Rx \in R holds. Find the smallest such bb, which works for all these functions .
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