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Poland - Second Round
1972 Poland - Second Round
1
ax^2 + bx + c = a(x - x_2)(x - x_3), 3x3 system
ax^2 + bx + c = a(x - x_2)(x - x_3), 3x3 system
Source: Polish MO Second Round 1972 p1
September 8, 2024
algebra
system of equations
System
Problem Statement
Prove that there are no real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
,
x
1
,
x
2
,
x
3
x_1, x_2, x_3
x
1
,
x
2
,
x
3
such that for every real number
x
x
x
a
x
2
+
b
x
+
c
=
a
(
x
−
x
2
)
(
x
−
x
3
)
ax^2 + bx + c = a(x - x_2)(x - x_3)
a
x
2
+
b
x
+
c
=
a
(
x
−
x
2
)
(
x
−
x
3
)
b
x
2
+
c
x
+
a
=
b
(
x
−
x
3
)
(
x
−
x
1
)
bx^2 + cx + a = b(x - x_3) (x - x_1)
b
x
2
+
c
x
+
a
=
b
(
x
−
x
3
)
(
x
−
x
1
)
c
x
2
+
a
x
+
b
=
c
(
x
−
x
1
)
(
x
−
x
2
)
cx^2 + ax + b = c(x - x_1) (x - x_2)
c
x
2
+
a
x
+
b
=
c
(
x
−
x
1
)
(
x
−
x
2
)
and
x
1
≠
x
2
x_1 \neq x_2
x
1
=
x
2
,
x
2
≠
x
3
x_2 \neq x_3
x
2
=
x
3
,
x
3
≠
x
1
x_3 \neq x_1
x
3
=
x
1
,
a
b
c
≠
0
abc \neq 0
ab
c
=
0
.
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