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1977 IMO Longlists
7
7
Part of
1977 IMO Longlists
Problems
(1)
Not quite Cauchy-Schwarz [ILL 1977]
Source:
1/11/2011
Prove the following assertion: If
c
1
,
c
2
,
…
,
c
n
(
n
≥
2
)
c_1,c_2,\ldots ,c_n\ (n\ge 2)
c
1
,
c
2
,
…
,
c
n
(
n
≥
2
)
are real numbers such that
(
n
−
1
)
(
c
1
2
+
c
2
2
+
⋯
+
c
n
2
)
=
(
c
1
+
c
2
+
⋯
+
c
n
)
2
,
(n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2,
(
n
−
1
)
(
c
1
2
+
c
2
2
+
⋯
+
c
n
2
)
=
(
c
1
+
c
2
+
⋯
+
c
n
)
2
,
then either all these numbers are nonnegative or all these numbers are nonpositive.
inequalities
algebra proposed
algebra