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Tournament Of Towns
1987 Tournament Of Towns
(154) 5
(154) 5
Part of
1987 Tournament Of Towns
Problems
(1)
TOT 154 1987 Autumn J5 A^2 + B^2 + C^2 \le 2(AB + BC + CA)
Source:
4/19/2020
We are given three non-negative numbers
A
,
B
A , B
A
,
B
and
C
C
C
about which it is known that
A
4
+
B
4
+
C
4
≤
2
(
A
2
B
2
+
B
2
C
2
+
C
2
A
2
)
A^4 + B^4 + C^4 \le 2(A^2B^2 + B^2C^2 + C^2A^2)
A
4
+
B
4
+
C
4
≤
2
(
A
2
B
2
+
B
2
C
2
+
C
2
A
2
)
(a) Prove that each of
A
,
B
A, B
A
,
B
and
C
C
C
is not greater than the sum of the others. (b) Prove that
A
2
+
B
2
+
C
2
≤
2
(
A
B
+
B
C
+
C
A
)
A^2 + B^2 + C^2 \le 2(AB + BC + CA)
A
2
+
B
2
+
C
2
≤
2
(
A
B
+
BC
+
C
A
)
. (c) Does the original inequality follow from the one in (b)?(V.A. Senderov , Moscow)
inequalities
algebra