MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
1966 German National Olympiad
4
4x4 system x_1x2 + x_1x_3 + x_2x_3 + x_4 = 2
4x4 system x_1x2 + x_1x_3 + x_2x_3 + x_4 = 2
Source: 1965-66 Germany R4 12.4
October 12, 2024
algebra
system of equations
System
Problem Statement
Determine all ordered quadruples of real numbers
(
x
1
,
x
2
,
x
3
,
x
4
)
(x_1, x_2, x_3, x_4)
(
x
1
,
x
2
,
x
3
,
x
4
)
for which the following system of equations exists, is fulfilled:
x
1
x
2
+
x
1
x
3
+
x
2
x
3
+
x
4
=
2
x_1x_2 + x_1x_3 + x_2x_3 + x_4 = 2
x
1
x
2
+
x
1
x
3
+
x
2
x
3
+
x
4
=
2
x
1
x
2
+
x
1
x
4
+
x
2
x
4
+
x
3
=
2
x_1x_2 + x_1x_4 + x_2x_4 + x_3 = 2
x
1
x
2
+
x
1
x
4
+
x
2
x
4
+
x
3
=
2
x
1
x
3
+
x
1
x
4
+
x
3
x
4
+
x
2
=
2
x_1x_3 + x_1x_4 + x_3x_4 + x_2 = 2
x
1
x
3
+
x
1
x
4
+
x
3
x
4
+
x
2
=
2
x
2
x
3
+
x
2
x
4
+
x
3
x
4
+
x
1
=
2
x_2x_3 + x_2x_4 + x_3x_4 + x_1 = 2
x
2
x
3
+
x
2
x
4
+
x
3
x
4
+
x
1
=
2
Back to Problems
View on AoPS