IMO 1969 LL BUL8

Find all functions f defined for all x that satisfy the condition

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IMO 1969 LL BUL8

Origin: BUL

Problem

Find all functions f defined for all x that satisfy the condition xf(y) + yf(x) = (x + y)f(x)f(y), for all x and y. Prove that exactly two of them are continuous.