Problem 1

Let

\mathbb{Z}

be the set of integers. Determine all functions

f\colon\mathbb{Z}\to\mathbb{Z}

such that, for all integers

a

and

b

f(2a) + 2f(b) = f(f(a + b)).

Step 2 of 7: Compare two substitutions

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In plain words

Two different pairs give two expressions for the same quantity f(f(x)).

P(0,x),P(x,0)\ \Longrightarrow\ f(2x)=2f(x)-f(0)

Detailed analysis

Comparing P(0, x) and P(x, 0), both equal f(f(x)): the first gives f(0) + 2f(x), and the second gives f(2x) + 2f(0). Setting these equal and rearranging shows f(2x) equals 2f(x) minus f(0).