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Israel Olympic Revenge
2021 Israel Olympic Revenge
4
a^2+b^2+c^2+abc=4 condition
a^2+b^2+c^2+abc=4 condition
Source: Israeli Olympic Revenge 2021, Problem 4
August 29, 2021
inequalities
Problem Statement
Prove that the inequality
4
a
+
b
c
+
4
+
4
b
+
c
a
+
4
+
4
c
+
a
b
+
4
≤
1
+
1
2
a
+
1
+
1
2
b
+
1
+
1
2
c
+
1
\frac{4}{a+bc+4}+\frac{4}{b+ca+4}+\frac{4}{c+ab+4}\le 1+\frac{1}{2a+1}+\frac{1}{2b+1}+\frac{1}{2c+1}
a
+
b
c
+
4
4
+
b
+
c
a
+
4
4
+
c
+
ab
+
4
4
≤
1
+
2
a
+
1
1
+
2
b
+
1
1
+
2
c
+
1
1
holds for all positive reals
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
2
+
b
2
+
c
2
+
a
b
c
=
4
a^2+b^2+c^2+abc=4
a
2
+
b
2
+
c
2
+
ab
c
=
4
.
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