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Russia Contests
Saint Petersburg Mathematical Olympiad
2014 Saint Petersburg Mathematical Olympiad
4
Interesting inequality
Interesting inequality
Source: St Petersburg Olympiad 2014, Grade 10, P4
October 26, 2017
algebra
inequalities
Problem Statement
a
1
≥
a
2
≥
.
.
.
≥
a
100
n
>
0
a_1\geq a_2\geq... \geq a_{100n}>0
a
1
≥
a
2
≥
...
≥
a
100
n
>
0
If we take from
(
a
1
,
a
2
,
.
.
.
,
a
100
n
)
(a_1,a_2,...,a_{100n})
(
a
1
,
a
2
,
...
,
a
100
n
)
some
2
n
+
1
2n+1
2
n
+
1
numbers
b
1
≥
b
2
≥
.
.
.
≥
b
2
n
+
1
b_1\geq b_2 \geq ... \geq b_{2n+1}
b
1
≥
b
2
≥
...
≥
b
2
n
+
1
then
b
1
+
.
.
.
+
b
n
>
b
n
+
1
+
.
.
.
b
2
n
+
1
b_1+...+b_n > b_{n+1}+...b_{2n+1}
b
1
+
...
+
b
n
>
b
n
+
1
+
...
b
2
n
+
1
Prove, that
(
n
+
1
)
(
a
1
+
.
.
.
+
a
n
)
>
a
n
+
1
+
a
n
+
2
+
.
.
.
+
a
100
n
(n+1)(a_1+...+a_n)>a_{n+1}+a_{n+2}+...+a_{100n}
(
n
+
1
)
(
a
1
+
...
+
a
n
)
>
a
n
+
1
+
a
n
+
2
+
...
+
a
100
n
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