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Omni-MATH / 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…

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problem

Two different integers uu and vv are written on a board. We perform a sequence of steps. At each step we do one of the following two operations: (i) If aa and bb are different integers on the board, then we can write a+ba + b on the board, if it is not already there. (ii) If aa, bb and cc are three different integers on the board, and if an integer xx satisfies ax2+bx+c=0ax^2 +bx+c = 0, then we can write xx on the board, if it is not already there. Determine all pairs of starting numbers (u,v)(u, v) from which any integer can eventually be written on the board after a finite sequence of steps.
Plain-text mathematical notation (without MathML)
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 finite sequence of steps.
Original LaTeX notation
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 $ax^2 +bx+c = 0$,
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 finite sequence of steps.

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Source and history

task_id
02331d68-582e-5c47-947e-89ec72cc2c9b
task_key
test--02331d68-582e-5c47-947e-89ec72cc2c9b
task_revision_id
1
config / split
/ test

Official source

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