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Moldova Contests
Moldova Team Selection Test
2017 Moldova Team Selection Test
6
6
Part of
2017 Moldova Team Selection Test
Problems
(1)
Basic inequality involving a condition [Moldova TST 2017, D2, P2]
Source: Moldova TST 2017, Day 2, Problem 2
3/19/2017
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers that satisfy
a
+
b
+
c
=
a
b
c
a+b+c=abc
a
+
b
+
c
=
ab
c
. Prove that
(
1
+
a
2
)
(
1
+
b
2
)
+
(
1
+
b
2
)
(
1
+
c
2
)
+
(
1
+
a
2
)
(
1
+
c
2
)
−
(
1
+
a
2
)
(
1
+
b
2
)
(
1
+
c
2
)
≥
4.
\sqrt{(1+a^2)(1+b^2)}+\sqrt{(1+b^2)(1+c^2)}+\sqrt{(1+a^2)(1+c^2)}-\sqrt{(1+a^2)(1+b^2)(1+c^2)} \ge 4.
(
1
+
a
2
)
(
1
+
b
2
)
+
(
1
+
b
2
)
(
1
+
c
2
)
+
(
1
+
a
2
)
(
1
+
c
2
)
−
(
1
+
a
2
)
(
1
+
b
2
)
(
1
+
c
2
)
≥
4.
inequalities
algebra