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This question, again, comprises two independent parts.
(i) Show that if

(k+1)

integers are chosen from

\left\{1,2,3,...,2k+1\right\}

, then among the chosen integers there are always two that are coprime.
(ii) Let

A=\left\{1,2,\ldots,n\right\}.

Prove that if

n>11

then there is a bijective map

f: A\to A

with the property that, for every

a\in A

, exactly one of

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

and

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

holds.

Problem Statement