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2019 JBMO Shortlist
A7
JBMO Shortlist 2019 A7
JBMO Shortlist 2019 A7
Source:
September 12, 2020
algebra
Problem Statement
Show that for any positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
such that
a
+
b
+
c
=
a
b
+
b
c
+
c
a
a + b + c = ab + bc + ca
a
+
b
+
c
=
ab
+
b
c
+
c
a
, the following inequality holds
3
+
a
3
+
1
2
3
+
b
3
+
1
2
3
+
c
3
+
1
2
3
≤
2
(
a
+
b
+
c
)
3 + \sqrt[3]{\frac{a^3+1}{2}}+\sqrt[3]{\frac{b^3+1}{2}}+\sqrt[3]{\frac{c^3+1}{2}}\leq 2(a+b+c)
3
+
3
2
a
3
+
1
+
3
2
b
3
+
1
+
3
2
c
3
+
1
≤
2
(
a
+
b
+
c
)
Proposed by Dorlir Ahmeti, Albania
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