Problem 1.  Two different integers u and v are written on a board. We perform a sequence of steps.At each step we do one of the following two operations:
(i) If a and b are different integers on the board, then we can write a + b on the board, if it is not already there.
(ii) If a, b and c are three different integers on the board, and if an integer x satisfies ax² + bx + c = 0, then we can write x on the board, if it is not already there.
Determine all pairs of starting numbers ( u, v ) from which any integer can eventually be written on the board after a fi nite sequence of steps.