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International Contests
EGMO
2024 EGMO
1
1
Part of
2024 EGMO
Problems
(1)
Operations with numbers on a board
Source: EGMO 2024, P1
4/13/2024
Two different integers
u
u
u
and
v
v
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
a
a
and
b
b
b
are different integers on the board, then we can write
a
+
b
a + b
a
+
b
on the board, if it is not already there. (ii) If
a
a
a
,
b
b
b
and
c
c
c
are three different integers on the board, and if an integer
x
x
x
satisfies
a
x
2
+
b
x
+
c
=
0
ax^2 +bx+c = 0
a
x
2
+
b
x
+
c
=
0
, then we can write
x
x
x
on the board, if it is not already there.Determine all pairs of starting numbers
(
u
,
v
)
(u, v)
(
u
,
v
)
from which any integer can eventually be written on the board after a finite sequence of steps.
EGMO