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Vietnam Team Selection Test
2011 Vietnam Team Selection Test
3
x_i real numbers with sum zero, n>=3
x_i real numbers with sum zero, n>=3
Source: Vietnamese TST 2011 P3
April 27, 2011
inequalities
inequalities proposed
Problem Statement
Let
n
n
n
be a positive integer
≥
3.
\geq 3.
≥
3.
There are
n
n
n
real numbers
x
1
,
x
2
,
⋯
x
n
x_1,x_2,\cdots x_n
x
1
,
x
2
,
⋯
x
n
that satisfy:
{
x
1
≥
x
2
≥
⋯
≥
x
n
;
x
1
+
x
2
+
⋯
+
x
n
=
0
;
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
n
(
n
−
1
)
.
\left\{\begin{aligned}&\ x_1\ge x_2\ge\cdots \ge x_n;\\& \ x_1+x_2+\cdots+x_n=0;\\& \ x_1^2+x_2^2+\cdots+x_n^2=n(n-1).\end{aligned}\right.
⎩
⎨
⎧
x
1
≥
x
2
≥
⋯
≥
x
n
;
x
1
+
x
2
+
⋯
+
x
n
=
0
;
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
n
(
n
−
1
)
.
Find the maximum and minimum value of the sum
S
=
x
1
+
x
2
.
S=x_1+x_2.
S
=
x
1
+
x
2
.
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