Shivang's function

Algebra Level 5

Consider the set of all functions f:QQ f:\mathbb{Q} \rightarrow \mathbb{Q} such that

f(x+f(y))=f(x+y)+f(y)x,yQ.f(x+f(y))=f(x+y)+f(y) \qquad \forall x, y \in \mathbb{Q}.

Find the sum of all possible (distinct) values of f(99) f(99) .

This problem is posed by Shivang J.

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