Consider all functions f:Z→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.
A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.
Solution
For almost all integers x,f(x)=−x−20. If f(x)=−x−20, then f(−x−20+2x+20)=15⟹−x−20=15⟹x=−35 Now it suffices to prove that the f(−35) can take any value. f(−35)=15 in the function f(x)≡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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