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1992 Poland - First Round
9
Sum of squares vs the sum squared
Sum of squares vs the sum squared
Source: Poland Math Olympiad 1992 Round 1 #9
June 2, 2023
inequalities
Problem Statement
Prove that for all real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
the inequality
(
a
2
+
b
2
−
c
2
)
(
b
2
+
c
2
−
a
2
)
(
c
2
+
a
2
−
b
2
)
≤
(
a
+
b
−
c
)
2
(
b
+
c
−
a
)
2
(
c
+
a
−
b
)
2
(a^2+b^2-c^2)(b^2+c^2-a^2)(c^2+a^2-b^2) \leq (a+b-c)^2(b+c-a)^2(c+a-b)^2
(
a
2
+
b
2
−
c
2
)
(
b
2
+
c
2
−
a
2
)
(
c
2
+
a
2
−
b
2
)
≤
(
a
+
b
−
c
)
2
(
b
+
c
−
a
)
2
(
c
+
a
−
b
)
2
holds.
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