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National and Regional Contests
India Contests
India National Olympiad
1996 India National Olympiad
3
Yet another system!
Yet another system!
Source: INMO 1996 Problem 3
October 6, 2005
algebra
Problem Statement
Solve the following system for real
a
,
b
,
c
,
d
,
e
a , b, c, d, e
a
,
b
,
c
,
d
,
e
:
{
3
a
=
(
b
+
c
+
d
)
3
3
b
=
(
c
+
d
+
e
)
3
3
c
=
(
d
+
e
+
a
)
3
3
d
=
(
e
+
a
+
b
)
3
3
e
=
(
a
+
b
+
c
)
3
.
\left\{ \begin{array}{ccc} 3a & = & ( b + c+ d)^3 \\ 3b & = & ( c + d +e ) ^3 \\ 3c & = & ( d + e +a )^3 \\ 3d & = & ( e + a +b )^3 \\ 3e &=& ( a + b +c)^3. \end{array}\right.
⎩
⎨
⎧
3
a
3
b
3
c
3
d
3
e
=
=
=
=
=
(
b
+
c
+
d
)
3
(
c
+
d
+
e
)
3
(
d
+
e
+
a
)
3
(
e
+
a
+
b
)
3
(
a
+
b
+
c
)
3
.
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