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MathLinks Contest 5th
3.3
0533 inequalities 5th edition Round 3 p3
0533 inequalities 5th edition Round 3 p3
Source:
May 6, 2021
inequalities
5th edition
algebra
Problem Statement
Let
x
1
,
x
2
,
.
.
.
x
n
x_1, x_2,... x_n
x
1
,
x
2
,
...
x
n
be positive numbers such that
S
=
x
1
+
x
2
+
.
.
.
+
x
n
=
1
x
1
+
.
.
.
+
1
x
n
S = x_1+x_2+...+x_n =\frac{1}{x_1}+...+\frac{1}{x_n}
S
=
x
1
+
x
2
+
...
+
x
n
=
x
1
1
+
...
+
x
n
1
Prove that
∑
i
=
1
n
1
n
−
1
+
x
i
≥
∑
i
=
1
n
1
1
+
S
−
x
i
\sum_{i=1}^{n} \frac{1}{n - 1 + x_i} \ge \sum_{i=1}^{n} \frac{1}{1+S - x_i}
i
=
1
∑
n
n
−
1
+
x
i
1
≥
i
=
1
∑
n
1
+
S
−
x
i
1
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