Problem 5

Determine all functions

f:\mathbb{Z}\to\mathbb{Z}

such that

f(f(a)-b)+b\,f(2a)

is a perfect square for all integers

a

and

b

Step 1 of 5: Answer: two families

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f(n)=n^2\ \text{for all }n;\qquad\text{or}\qquad f(n)=0\ (n\text{ even}),\ f(n)=\text{any perfect square}\ (n\text{ odd})

Detailed analysis

The solutions are $f(n)=n^2$ for all $n$ , together with the family where $f(n)=0$ for every even $n$ and $f(n)$ is an arbitrary perfect square (chosen independently for each odd $n$ ).