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Additive or multiplicative functional equation f(x + y)

This type of question, asked 3 times: 2021 Paper 23, 30 Jan 2023, 9 Apr 2024. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 23

    65
    Let f:N→Nf:N \rightarrow N be a function such that f(m+n)=f(m)+f(n)f(m+n) =f(m) +f(n) for every m,n∈Nm,n\in N. If f(6)=18f(6) = 18, then f(2)⋅f(3)f(2) \cdot f(3) is equal to:

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  2. Question 2

    30 Jan 2023

    80
    Suppose f:R→(0,∞)f:R \rightarrow (0,\infty) be a differentiable function such that 5f(x+y)=f(x)⋅f(y),∀x,y∈R5f(x + y) = f(x) \cdot f(y),\forall x,y \in R. If f(3)=320f(3) = 320, then ∑n=05f(n)\sum_{n = 0}^{5} f(n) is equal to:

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  3. Question 3

    9 Apr 2024

    89
    If a function ff satisfies f(m+n)=f(m)+f(n)f(m + n) = f(m) + f(n) for all m,n∈Nm,n \in N and f(1)=1f(1) = 1, then the largest natural number λ\lambda such that ∑k=12022f(λ+k)≤(2022)2\sum_{k = 1}^{2022} f(\lambda + k) \leq (2022)^{2} is equal to