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4
sum x_i/(1+(n-1)x_i) <=1 where 0 < ,x_i <= 1
sum x_i/(1+(n-1)x_i) <=1 where 0 < ,x_i <= 1
Source: Singapore Open Math Olympiad 2004 2nd Round p4 SMO
April 3, 2020
inequalities
algebra
Problem Statement
If
0
<
x
1
,
x
2
,
.
.
.
,
x
n
≤
1
0 <x_1,x_2,...,x_n\le 1
0
<
x
1
,
x
2
,
...
,
x
n
≤
1
, where
n
≥
1
n \ge 1
n
≥
1
, show that
x
1
1
+
(
n
−
1
)
x
1
+
x
2
1
+
(
n
−
1
)
x
2
+
.
.
.
+
x
n
1
+
(
n
−
1
)
x
n
≤
1
\frac{x_1}{1+(n-1)x_1}+\frac{x_2}{1+(n-1)x_2}+...+\frac{x_n}{1+(n-1)x_n}\le 1
1
+
(
n
−
1
)
x
1
x
1
+
1
+
(
n
−
1
)
x
2
x
2
+
...
+
1
+
(
n
−
1
)
x
n
x
n
≤
1
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