2015 HMMT November General Round Problem 6

Consider all functions f : \mathbb{Z} \to \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.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2015 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.