MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
JBMO TST - Turkey
2021 JBMO TST - Turkey
7
7
Part of
2021 JBMO TST - Turkey
Problems
(1)
Finite algebraic moves on a blackboard
Source: 2021 Turkey JBMO TST P7
5/24/2021
Initially on a blackboard, the equation
a
1
x
2
+
b
1
x
+
c
=
0
a_1x^2+b_1x+c=0
a
1
x
2
+
b
1
x
+
c
=
0
is written where
a
1
,
b
1
,
c
1
a_1, b_1, c_1
a
1
,
b
1
,
c
1
are integers and
(
a
1
+
c
1
)
b
1
>
0
(a_1+c_1)b_1 > 0
(
a
1
+
c
1
)
b
1
>
0
. At each move, if the equation
a
x
2
+
b
x
+
c
=
0
ax^2+bx+c=0
a
x
2
+
b
x
+
c
=
0
is written on the board and there is a
x
∈
R
x \in \mathbb{R}
x
∈
R
satisfying the equation, Alice turns this equation into
(
b
+
c
)
x
2
+
(
c
+
a
)
x
+
(
a
+
b
)
=
0
(b+c)x^2+(c+a)x+(a+b)=0
(
b
+
c
)
x
2
+
(
c
+
a
)
x
+
(
a
+
b
)
=
0
. Prove that Alice will stop after a finite number of moves.
algebra
algebra proposed