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Smallest n....Rationals
Smallest n....Rationals
Source:
December 9, 2010
algebra unsolved
algebra
Problem Statement
Determine the smallest odd integer
n
≥
3
n \ge 3
n
≥
3
, for which there exist
n
n
n
rational numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1 , x_2 , . . . , x_n
x
1
,
x
2
,
...
,
x
n
with the following properties:
(
a
)
(a)
(
a
)
∑
i
=
1
n
x
i
=
0
,
∑
i
=
1
n
x
i
2
=
1.
\sum_{i=1}^{n} x_i =0 , \ \sum_{i=1}^{n} x_i^2 = 1.
i
=
1
∑
n
x
i
=
0
,
i
=
1
∑
n
x
i
2
=
1.
(
b
)
(b)
(
b
)
x
i
⋅
x
j
≥
−
1
n
∀
1
≤
i
,
j
≤
n
.
x_i \cdot x_j \ge - \frac 1n \ \forall \ 1 \le i,j \le n.
x
i
⋅
x
j
≥
−
n
1
∀
1
≤
i
,
j
≤
n
.
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