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Baltic Way
2010 Baltic Way
1
Symmetric simultaneous equations of degree 2010
Symmetric simultaneous equations of degree 2010
Source: Baltic Way 2010
November 19, 2010
algebra
system of equations
algebra proposed
Problem Statement
Find all quadruples of real numbers
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
satisfying the system of equations
{
(
b
+
c
+
d
)
2010
=
3
a
(
a
+
c
+
d
)
2010
=
3
b
(
a
+
b
+
d
)
2010
=
3
c
(
a
+
b
+
c
)
2010
=
3
d
\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}
⎩
⎨
⎧
(
b
+
c
+
d
)
2010
=
3
a
(
a
+
c
+
d
)
2010
=
3
b
(
a
+
b
+
d
)
2010
=
3
c
(
a
+
b
+
c
)
2010
=
3
d
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