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exists g(g(a)) = f(a) if f(m) = 1, f(m+n) = m+1 and f(i) = i+1 for all other i

Source: 1995 Bulgaria NMO, Round 4, p5

July 30, 2021

functionfunctionalalgebra

Problem Statement

Let

A = \{1,2,...,m + n\}

, where

m,n

are positive integers, and let the function f :

A \to A

be defined by:

f(m) = 1

,

f(m+n) = m+1

and

f(i) = i+1

for all the other

i

. (a) Prove that if

m

and

n

are odd, then there exists a function

g : A \to A

such that

g(g(a)) = f(a)

for all

a \in A

. (b) Prove that if

m

is even, then there is a function

g : A\to A

such that

g(g(a))=f(a)

for all

a \in A

is and only if

n = m

.

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