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2019 Taiwan APMO Preliminary P6

Source: 2019 Taiwan APMO Preliminary

August 23, 2020
algebrafunctional equation

Problem Statement

Let N\mathbb{N} denote the set of all positive integers.Function f:N∪0→N∪0f:\mathbb{N}\cup{0}\rightarrow\mathbb{N}\cup{0} satisfies :for any two distinct positive integer a,ba,b, we have f(a)+f(b)−f(a+b)=2019f(a)+f(b)-f(a+b)=2019 (1)Find f(0)f(0) (2)Let a1,a2,...,a100a_1,a_2,...,a_{100} be 100 positive integers (they are pairwise distinct), find f(a1)+f(a2)+...+f(a100)−f(a1+a2+...+a100)f(a_1)+f(a_2)+...+f(a_{100})-f(a_1+a_2+...+a_{100})
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