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JBMO TST - Moldova
2020 Junior Balkan Team Selection Tests - Moldova
2
JBMO TST Moldova Problem 2
JBMO TST Moldova Problem 2
Source: JBMO TST Moldova Problem 2
October 13, 2020
algebra
Inequality
3-variable inequality
Moldova
Problem Statement
The positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy the equation
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove the identity:
(
a
+
b
c
)
(
b
+
c
a
)
c
+
a
b
+
(
b
+
c
a
)
(
c
+
a
b
)
a
+
b
c
+
(
c
+
a
b
)
(
a
+
b
c
)
b
+
c
a
=
2
\sqrt{\frac{(a+bc)(b+ca)}{c+ab}}+\sqrt{\frac{(b+ca)(c+ab)}{a+bc}}+\sqrt{\frac{(c+ab)(a+bc)}{b+ca}} = 2
c
+
ab
(
a
+
b
c
)
(
b
+
c
a
)
+
a
+
b
c
(
b
+
c
a
)
(
c
+
ab
)
+
b
+
c
a
(
c
+
ab
)
(
a
+
b
c
)
=
2
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