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China National Olympiad
1986 China National Olympiad
1
1
Part of
1986 China National Olympiad
Problems
(1)
China Mathematical Olympiad 1986 problem1
Source: China Mathematical Olympiad 1986 problem1
1/16/2014
We are given
n
n
n
reals
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots , a_n
a
1
,
a
2
,
⋯
,
a
n
such that the sum of any two of them is non-negative. Prove that the following statement and its converse are both true: if
n
n
n
non-negative reals
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots ,x_n
x
1
,
x
2
,
⋯
,
x
n
satisfy
x
1
+
x
2
+
⋯
+
x
n
=
1
x_1+x_2+\cdots +x_n=1
x
1
+
x
2
+
⋯
+
x
n
=
1
, then the inequality
a
1
x
1
+
a
2
x
2
+
⋯
+
a
n
x
n
≥
a
1
x
1
2
+
a
2
x
2
2
+
⋯
+
a
n
x
n
2
a_1x_1+a_2x_2+\cdots +a_nx_n\ge a_1x^2_1+ a_2x^2_2+\cdots + a_nx^2_n
a
1
x
1
+
a
2
x
2
+
⋯
+
a
n
x
n
≥
a
1
x
1
2
+
a
2
x
2
2
+
⋯
+
a
n
x
n
2
holds.
inequalities