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JBMO TST - Turkey
2013 JBMO TST - Turkey
4
4
Part of
2013 JBMO TST - Turkey
Problems
(1)
Inequality with three variables
Source: Turkey JBMO Team Selection Test 2013, P4
5/31/2013
For all positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfying
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
, prove that
a
4
+
5
b
4
a
(
a
+
2
b
)
+
b
4
+
5
c
4
b
(
b
+
2
c
)
+
c
4
+
5
a
4
c
(
c
+
2
a
)
≥
1
−
a
b
−
b
c
−
c
a
\frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca
a
(
a
+
2
b
)
a
4
+
5
b
4
+
b
(
b
+
2
c
)
b
4
+
5
c
4
+
c
(
c
+
2
a
)
c
4
+
5
a
4
≥
1
−
ab
−
b
c
−
c
a
inequalities
quadratics
algebra
polynomial
inequalities proposed