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2022 Israel TST
2
Real variable inequality, cyc sum a^5+a^3c^2
Real variable inequality, cyc sum a^5+a^3c^2
Source: 2022 Israel TST 3 P2
May 22, 2022
inequalities
Problem Statement
The numbers
a
a
a
,
b
b
b
, and
c
c
c
are real. Prove that
(
a
5
+
b
5
+
c
5
+
a
3
c
2
+
b
3
a
2
+
c
3
b
2
)
2
≥
4
(
a
2
+
b
2
+
c
2
)
(
a
5
b
3
+
b
5
c
3
+
c
5
a
3
)
(a^5+b^5+c^5+a^3c^2+b^3a^2+c^3b^2)^2\geq 4(a^2+b^2+c^2)(a^5b^3+b^5c^3+c^5a^3)
(
a
5
+
b
5
+
c
5
+
a
3
c
2
+
b
3
a
2
+
c
3
b
2
)
2
≥
4
(
a
2
+
b
2
+
c
2
)
(
a
5
b
3
+
b
5
c
3
+
c
5
a
3
)
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