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National and Regional Contests
Russia Contests
All-Russian Olympiad
1978 All Soviet Union Mathematical Olympiad
267
267
Part of
1978 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 267 All Soviet Union MO 1978, Σ(a_n - b_n)^2
Source:
7/11/2019
Given
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ... , a_n
a
1
,
a
2
,
...
,
a
n
. Define
b
k
=
a
1
+
a
2
+
.
.
.
+
a
k
k
b_k = \frac{a_1 + a_2 + ... + a_k}{k}
b
k
=
k
a
1
+
a
2
+
...
+
a
k
for
1
≤
k
≤
n
.
1 \le k\le n.
1
≤
k
≤
n
.
Let
C
=
(
a
1
−
b
1
)
2
+
(
a
2
−
b
2
)
2
+
.
.
.
+
(
a
n
−
b
n
)
2
,
D
=
(
a
1
−
b
n
)
2
+
(
a
2
−
b
n
)
2
+
.
.
.
+
(
a
n
−
b
n
)
2
C = (a_1 - b_1)^2 + (a_2 - b_2)^2 + ... + (a_n - b_n)^2, D = (a_1 - b_n)^2 + (a_2 - b_n)^2 + ... + (a_n - b_n)^2
C
=
(
a
1
−
b
1
)
2
+
(
a
2
−
b
2
)
2
+
...
+
(
a
n
−
b
n
)
2
,
D
=
(
a
1
−
b
n
)
2
+
(
a
2
−
b
n
)
2
+
...
+
(
a
n
−
b
n
)
2
Prove that
C
≤
D
≤
2
C
C \le D \le 2C
C
≤
D
≤
2
C
.
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