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Problems
Contests
International Contests
Pan African
2001 Pan African
1
Equation
Equation
Source: Pan African 2001
October 4, 2005
Problem Statement
Let
n
n
n
be a positive integer, and let
a
>
0
a>0
a
>
0
be a real number. Consider the equation:
∑
i
=
1
n
(
x
i
2
+
(
a
−
x
i
)
2
)
=
n
a
2
\sum_{i=1}^{n}(x_i^2+(a-x_i)^2)= na^2
i
=
1
∑
n
(
x
i
2
+
(
a
−
x
i
)
2
)
=
n
a
2
How many solutions (
x
1
,
x
2
⋯
,
x
n
x_1, x_2 \cdots , x_n
x
1
,
x
2
⋯
,
x
n
) does this equation have, such that:
0
≤
x
i
≤
a
,
i
∈
N
+
0 \leq x_i \leq a, i \in N^+
0
≤
x
i
≤
a
,
i
∈
N
+
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