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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1969 Swedish Mathematical Competition
3
3
Part of
1969 Swedish Mathematical Competition
Problems
(1)
sum a_ib_i <= sum a_i B_i
Source: 1969 Swedish Mathematical Competition p3
3/21/2021
a
1
≤
a
2
≤
.
.
.
≤
a
n
a_1 \le a_2 \le ... \le a_n
a
1
≤
a
2
≤
...
≤
a
n
is a sequence of reals
b
1
,
b
2
,
b
3
,
.
.
.
,
b
n
b_1, _b2, b_3,..., b_n
b
1
,
b
2
,
b
3
,
...
,
b
n
is any rearrangement of the sequence
B
1
≤
B
2
≤
.
.
.
≤
B
n
B_1 \le B_2 \le ...\le B_n
B
1
≤
B
2
≤
...
≤
B
n
. Show that
∑
a
i
b
i
≤
∑
a
i
B
i
\sum a_ib_i \le \sum a_i B_i
∑
a
i
b
i
≤
∑
a
i
B
i
.
algebra
inequalities