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Cyprus JBMO TST
2022 Cyprus JBMO TST
3
Two similar looking inequalities with the constraint abc=1
Two similar looking inequalities with the constraint abc=1
Source: Cyprus 2022 Junior TST-3 Problem 3
March 26, 2022
inequalities
AM-GM
Cauchy-Schwarz inequality
Problem Statement
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers with
a
b
c
=
1
abc=1
ab
c
=
1
, prove that (a)
2
(
a
b
a
+
b
+
b
c
b
+
c
+
c
a
c
+
a
)
⩾
9
a
b
+
b
c
+
c
a
2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{ab+bc+ca}
2
(
a
+
b
ab
+
b
+
c
b
c
+
c
+
a
c
a
)
⩾
ab
+
b
c
+
c
a
9
(b)
2
(
a
b
a
+
b
+
b
c
b
+
c
+
c
a
c
+
a
)
⩾
9
a
2
b
+
b
2
c
+
c
2
a
2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{a^2 b+b^2 c+c^2 a}
2
(
a
+
b
ab
+
b
+
c
b
c
+
c
+
a
c
a
)
⩾
a
2
b
+
b
2
c
+
c
2
a
9
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