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Poland Contests
Polish MO Finals
1964 Polish MO Finals
2
sum a_i * sum b_y < n sum a_ib_i
sum a_i * sum b_y < n sum a_ib_i
Source: Polish MO Finals 1964 p2
August 30, 2024
algebra
inequalities
Problem Statement
Prove that if
a
1
<
a
2
<
…
<
a
n
a_1 < a_2 < \ldots < a_n
a
1
<
a
2
<
…
<
a
n
and
b
1
<
b
2
<
…
<
b
n
b_1 < b_2 < \ldots < b_n
b
1
<
b
2
<
…
<
b
n
, where
n
≥
2
n \geq 2
n
≥
2
, then
(
a
1
+
a
2
+
…
+
a
n
)
(
b
1
+
b
2
+
…
+
b
n
)
<
n
(
a
1
b
1
+
a
2
b
2
+
…
+
a
n
b
n
)
.
\qquad (a_1 + a_2 + \ldots + a_n)(b_1 + b_2 + \ldots + b_n) < n(a_1b_1 + a_2b_2 + \ldots + a_nb_n).
(
a
1
+
a
2
+
…
+
a
n
)
(
b
1
+
b
2
+
…
+
b
n
)
<
n
(
a
1
b
1
+
a
2
b
2
+
…
+
a
n
b
n
)
.
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