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Bijection f satisfying |i-j| <= k -> |f(i) - f(j)|<=k

Source: Serbian National Olympiad 2013, Problem 1

April 8, 2013

algebra proposedalgebra

Problem Statement

Let

k

be a natural number. Bijection

f:\mathbb{Z} \rightarrow \mathbb{Z}

has the following property: for any integers

i

and

j

|i-j|\leq k

implies

|f(i) - f(j)|\leq k

. Prove that for every

i,j\in \mathbb{Z}

it stands:

|f(i)-f(j)|= |i-j|.