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n=(\sum ai^2)(\sum bi^2)-(\sum aibi)^2 - OIMU 2008 Problem 5
n=(\sum ai^2)(\sum bi^2)-(\sum aibi)^2 - OIMU 2008 Problem 5
Source:
August 28, 2010
inequalities
number theory proposed
number theory
Problem Statement
Find all positive integers
n
n
n
such that there are positive integers
a
1
,
⋯
,
a
n
,
b
1
,
⋯
,
b
n
a_1,\cdots,a_n, b_1,\cdots,b_n
a
1
,
⋯
,
a
n
,
b
1
,
⋯
,
b
n
that satisfy
(
a
1
2
+
⋯
+
a
n
2
)
(
b
1
2
+
⋯
+
b
n
2
)
−
(
a
1
b
1
+
⋯
+
a
n
b
n
)
2
=
n
(a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2)-(a_1b_1+\cdots+a_nb_n)^2=n
(
a
1
2
+
⋯
+
a
n
2
)
(
b
1
2
+
⋯
+
b
n
2
)
−
(
a
1
b
1
+
⋯
+
a
n
b
n
)
2
=
n
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