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Rare Netherlands FE

Source: 2019 ISL A7

September 22, 2020
algebraIMO ShortlistIMO Shortlist 2019functional equationarithmetic sequence

Problem Statement

Let Z\mathbb Z be the set of integers. We consider functions f:Z→Zf :\mathbb Z\to\mathbb Z satisfying f(f(x+y)+y)=f(f(x)+y)f\left(f(x+y)+y\right)=f\left(f(x)+y\right) for all integers xx and yy. For such a function, we say that an integer vv is f-rare if the set Xv={x∈Z:f(x)=v}X_v=\{x\in\mathbb Z:f(x)=v\} is finite and nonempty. (a) Prove that there exists such a function ff for which there is an ff-rare integer. (b) Prove that no such function ff can have more than one ff-rare integer.
Netherlands
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