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National and Regional Contests
China Contests
China Northern MO
2012 China Northern MO
2
max sum x_i/n if sum x_i^2=111
max sum x_i/n if sum x_i^2=111
Source: China Northern MO 2012 p2 CNMO
May 4, 2024
algebra
inequalities
Problem Statement
Positive integers
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
(
n
∈
N
+
n \in N_+
n
∈
N
+
) satisfy
x
1
2
+
x
2
2
+
.
.
.
+
x
n
2
=
111
x_1^2 +x_2^2+...+x_n^2=111
x
1
2
+
x
2
2
+
...
+
x
n
2
=
111
, find the maximum possible value of
S
=
x
1
+
x
2
+
.
.
.
+
x
n
n
S =\frac{x_1 +x_2+...+x_n}{n}
S
=
n
x
1
+
x
2
+
...
+
x
n
.
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