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National and Regional Contests
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Canada National Olympiad
2016 Canada National Olympiad
2
System of Equations
System of Equations
Source: 2016 CMO #2
April 10, 2016
algebra
system of equations
Problem Statement
Consider the following system of
10
10
10
equations in
10
10
10
real variables
v
1
,
…
,
v
10
v_1, \ldots, v_{10}
v
1
,
…
,
v
10
:
v
i
=
1
+
6
v
i
2
v
1
2
+
v
2
2
+
⋯
+
v
10
2
(
i
=
1
,
…
,
10
)
.
v_i = 1 + \frac{6v_i^2}{v_1^2 + v_2^2 + \cdots + v_{10}^2} \qquad (i = 1, \ldots, 10).
v
i
=
1
+
v
1
2
+
v
2
2
+
⋯
+
v
10
2
6
v
i
2
(
i
=
1
,
…
,
10
)
.
Find all
10
10
10
-tuples
(
v
1
,
v
2
,
…
,
v
10
)
(v_1, v_2, \ldots , v_{10})
(
v
1
,
v
2
,
…
,
v
10
)
that are solutions of this system.
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