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Max and min of a function involving [x]

Source: Canada 1996

March 4, 2006
functionLaTeXnumber theory unsolvednumber theory

Problem Statement

Let r1r_1, r2r_2, …\ldots, rmr_m be a given set of mm positive rational numbers such that ∑k=1mrk=1\sum_{k=1}^m r_k = 1. Define the function ff by f(n)=n−∑k=1m [rkn]f(n)= n-\sum_{k=1}^m \: [r_k n] for each positive integer nn. Determine the minimum and maximum values of f(n)f(n). Here  [x]{\ [ x ]} denotes the greatest integer less than or equal to xx.
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