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Turkey Team Selection Test
2007 Turkey Team Selection Test
3
a+b+c=1
a+b+c=1
Source: turkey 2007 TST
April 10, 2007
inequalities
function
inequalities proposed
algebra
highschoolmath
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive reals such that their sum is
1
1
1
. Prove that
1
a
b
+
2
c
2
+
2
c
+
1
b
c
+
2
a
2
+
2
a
+
1
a
c
+
2
b
2
+
2
b
≥
1
a
b
+
b
c
+
a
c
.
\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.
ab
+
2
c
2
+
2
c
1
+
b
c
+
2
a
2
+
2
a
1
+
a
c
+
2
b
2
+
2
b
1
≥
ab
+
b
c
+
a
c
1
.
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