HMMT November, 2015

Algebra Difficulty 5.0 AIME Prove it United States

Problem:

Consider all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying
$f(f(x)+2x+20)=15$
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 $x$ .

Solution

Solution:

Answer: $-35$

For almost all integers $x$ , $f(x) \neq -x-20$ . If $f(x) = -x-20$ , then
$f(-x-20+2x+20) = 15 \Longrightarrow -x-20 = 15 \Longrightarrow x = -35$
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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