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2018 Indonesia MO
4
4
Part of
2018 Indonesia MO
Problems
(1)
Real game with a computer
Source: Indonesian National Science Olympiad 2018, Mathematics P4
7/6/2018
In a game, Andi and a computer take turns. At the beginning, the computer shows a polynomial
x
2
+
m
x
+
n
x^2 + mx + n
x
2
+
m
x
+
n
where
m
,
n
ā
Z
m,n \in \mathbb{Z}
m
,
n
ā
Z
, such that it doesn't have real roots. Andi then begins the game. On his turn, Andi may change a polynomial in the form
x
2
+
a
x
+
b
x^2 + ax + b
x
2
+
a
x
+
b
into either
x
2
+
(
a
+
b
)
x
+
b
x^2 + (a+b)x + b
x
2
+
(
a
+
b
)
x
+
b
or
x
2
+
a
x
+
(
a
+
b
)
x^2 + ax + (a+b)
x
2
+
a
x
+
(
a
+
b
)
. However, Andi may only choose a polynomial that has real roots. On the computer's turn, it simply switches the coefficient of
x
x
x
and the constant of the polynomial. Andi loses if he can't continue to play. Find all
(
m
,
n
)
(m,n)
(
m
,
n
)
such that Andi always loses (in finitely many turns).
algebra