Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME Find the answer

Consider all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $f(f(x)+2 x+20)=15$ Call an integer $n$ good if $f(n)$ can take any integer value. In other words, if we fix $n$ , for any integer $m$ , there exists a function $f$ such that $f(n)=m$ . Find the sum of all good integers $x$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

For almost all integers $x, f(x) \neq-x-20$ . If $f(x)=-x-20$ , then $f(-x-20+2 x+20)=15 \Longrightarrow-x-20=15 \Longrightarrow x=-35$ Now it suffices to prove that the $f(-35)$ can take any value. $f(-35)=15$ in the function $f(x) \equiv 15$ . Otherwise, set $f(-35)=c$ , and $f(x)=15$ for all other $x$ . It is easy to check that these functions all work.

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