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Macedonian Team Selection Test
2013 Macedonian Team Selection Test
Problem 4
Nonsymmetric inequality for sum of three fractions
Nonsymmetric inequality for sum of three fractions
Source: Macedonian TST for IMO 2013 - P1 day 2
March 28, 2021
inequalities
Problem Statement
Let
a
>
0
,
b
>
0
,
c
>
0
a>0,b>0,c>0
a
>
0
,
b
>
0
,
c
>
0
and
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Show the inequality
a
4
+
b
4
a
2
+
b
2
+
b
3
+
c
3
b
+
c
+
2
a
2
+
b
2
+
2
c
2
2
≥
1
2
\frac{a^4+b^4}{a^2+b^2}+\frac{b^3+c^3}{b+c} + \frac{2a^2+b^2+2c^2}{2} \geq \frac{1}{2}
a
2
+
b
2
a
4
+
b
4
+
b
+
c
b
3
+
c
3
+
2
2
a
2
+
b
2
+
2
c
2
≥
2
1
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