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    OlympiadMediumNumber Theory10–12

    Functional Equations

    Equações Funcionais

    Find all functions f:Z→Zf: \mathbb{Z} \to \mathbb{Z} such that f(x+y)=f(x)+f(y)f(x + y) = f(x) + f(y) for all integers x,yx, y, and f(1)=3f(1) = 3.

    First find f(0)f(0), f(2)f(2), f(3)f(3), etc. to discover the pattern.

    Solution

    Step 1 of 7

    1. 1.f(0)=f(0+0)=f(0)+f(0)f(0) = f(0+0) = f(0)+f(0) → f(0)=0f(0) = 0