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Does there exist positive integers m,n such that f(m,n)=2018?

Source: 2018 China Southeast MO Grade 11 P7

July 31, 2018
number theory

Problem Statement

For positive integers m,n,m,n, define f(m,n)f(m,n) as the number of ordered triples (x,y,z)(x,y,z) of integers such that {xyz=x+y+z+m,max⁡{∣x∣,∣y∣,∣z∣}≤n \begin{cases} xyz=x+y+z+m, \\ \max\{|x|,|y|,|z|\} \leq n \end{cases} Does there exist positive integers m,n,m,n, such that f(m,n)=2018?f(m,n)=2018? Please prove your conclusion.
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