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2020 PUMaC Algebra A3 / B5

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January 1, 2022
floor functionalgebra

Problem Statement

Let {x}=x−⌊x⌋\{x\} = x- \lfloor x \rfloor . Consider a function f from the set {1,2,...,2020}\{1, 2, . . . , 2020\} to the half-open interval [0,1)[0, 1). Suppose that for all x,y,x, y, there exists a zz so that {f(x)+f(y)}=f(z)\{f(x) + f(y)\} = f(z). We say that a pair of integers m,nm, n is valid if 1≤m,n≤20201 \le m, n \le 2020 and there exists a function ff satisfying the above so f(1)=mnf(1) = \frac{m}{n}. Determine the sum over all valid pairs m,nm, n of mn{m}{n}.
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