Maths Olympiad Prep

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Algebra Difficulty 4.4 AIME Find the answer United States

Problem:
Let the function f:Z→Zf: \mathbb{Z} \rightarrow \mathbb{Z} take only integer inputs and have integer outputs. For any integers xx and yy, ff satisfies
f(x)+f(y)=f(x+1)+f(y−1) f(x)+f(y)=f(x+1)+f(y-1)
If f(2016)=6102f(2016)=6102 and f(6102)=2016f(6102)=2016, what is f(1)f(1)?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
We have
f(x+1)=f(x)+f(y)−f(y−1) f(x+1)=f(x)+f(y)-f(y-1)
If yy is fixed, we have
f(x+1)=f(x)+ constant  f(x+1)=f(x)+\text{ constant }
implying ff is linear. Using our two points, then, we get f(x)=8118−xf(x)=8118-x, so f(1)=8117f(1)=8117

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.