# Think periodically

Algebra Level 5

Consider the functions $$f(x)$$ such that $f(x+2)+f(x)+f(x-2)=f(x+1)+f(x-1)$ for all real $$x$$. Find the smallest positive number $$P$$ that is a period for all such functions $$f(x).$$

For extra credit, give an example of a non-zero function $$f(x)$$ of this form that has a fundamental period less than $$P.$$

Inspiration

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