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1
((a+b)^2 +1)/(c^2+2)+((b+c)^2 +1)/(a^2+2)+((c+a)^2 +1)/(b^2+2) >=3
((a+b)^2 +1)/(c^2+2)+((b+c)^2 +1)/(a^2+2)+((c+a)^2 +1)/(b^2+2) >=3
Source: 2010 Romania JBMO TST 5.3
June 3, 2020
inequalities
algebra
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be real numbers with the property as
a
b
+
b
c
+
c
a
=
1
ab + bc + ca = 1
ab
+
b
c
+
c
a
=
1
. Show that:
(
a
+
b
)
2
+
1
c
2
+
2
+
(
b
+
c
)
2
+
1
a
2
+
2
+
(
c
+
a
)
2
+
1
b
2
+
2
≥
3
\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3
c
2
+
2
(
a
+
b
)
2
+
1
+
a
2
+
2
(
b
+
c
)
2
+
1
+
b
2
+
2
(
c
+
a
)
2
+
1
≥
3
.
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