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Eotvos Mathematical Competition (Hungary)
1899 Eotvos Mathematical Competition
2
Let $x_1$ and $x_2$ be the roots of the equation $$x^2-(a+d)x+ad-bc=0.$$ Show th
Let $x_1$ and $x_2$ be the roots of the equation $$x^2-(a+d)x+ad-bc=0.$$ Show th
Source: Eotvos 1899 p2
March 15, 2020
algebra
Problem Statement
Let
x
1
x_1
x
1
and
x
2
x_2
x
2
be the roots of the equation
x
2
−
(
a
+
d
)
x
+
a
d
−
b
c
=
0.
x^2-(a+d)x+ad-bc=0.
x
2
−
(
a
+
d
)
x
+
a
d
−
b
c
=
0.
Show that
x
1
3
x^3_1
x
1
3
and
x
2
3
x^3_2
x
2
3
are the roots of
y
3
−
(
a
3
+
d
3
+
3
a
b
c
+
3
b
c
d
)
y
+
(
a
d
−
b
c
)
3
=
0.
y^3-(a^3+d^3+3abc+3bcd)y+(ad-bc)^3 =0.
y
3
−
(
a
3
+
d
3
+
3
ab
c
+
3
b
c
d
)
y
+
(
a
d
−
b
c
)
3
=
0.
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