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function f has property P, f(x) = ax + b

Source: 2003 Romania District VIII p2

August 15, 2024
algebra

Problem Statement

Let M⊂RM \subset R be a finite set containing at least two elements. We say that the function ff has property PP if f:M→Mf : M \to M and there are a∈R∗a \in R^* and b∈Rb \in R such that f(x)=ax+bf(x) = ax + b. (a) Show that there is at least a function having property PP. (b) Show that there are at most two functions having property PP. (c) If MM has 20032003 elements with sum 00 and if there are two functions with property PP, prove that 0∈M0 \in M.
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