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Moldova Contests
Moldova Team Selection Test
2021 Moldova Team Selection Test
7
7
Part of
2021 Moldova Team Selection Test
Problems
(1)
\frac{a+1}{\sqrt{a+bc}}+\frac{b+1}{\sqrt{b+ca}}+\frac{c+1}{\sqrt{c+ab}}
Source: Moldova TST 2021
9/20/2021
Positive real numbers
a
a
a
,
b
b
b
,
c
c
c
satisfy
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Show that
a
+
1
a
+
b
c
+
b
+
1
b
+
c
a
+
c
+
1
c
+
a
b
≥
2
a
2
+
b
2
+
c
2
.
\frac{a+1}{\sqrt{a+bc}}+\frac{b+1}{\sqrt{b+ca}}+\frac{c+1}{\sqrt{c+ab}} \geq \frac{2}{a^2+b^2+c^2}.
a
+
b
c
a
+
1
+
b
+
c
a
b
+
1
+
c
+
ab
c
+
1
≥
a
2
+
b
2
+
c
2
2
.
When does the equality take place?
inequalities