MathDB

A3

Part of 2017 IMO Shortlist

Problems(1)

Show that f is bijective on T

Source: IMO Shortlist 2017 A3

7/10/2018
Let SS be a finite set, and let A\mathcal{A} be the set of all functions from SS to SS. Let ff be an element of A\mathcal{A}, and let T=f(S)T=f(S) be the image of SS under ff. Suppose that f∘g∘f≠g∘f∘gf\circ g\circ f\ne g\circ f\circ g for every gg in A\mathcal{A} with g≠fg\ne f. Show that f(T)=Tf(T)=T.
IMO Shortlistfunctionalgebra