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Problems
Contests
National and Regional Contests
Latvia Contests
Latvia BW TST
2022 Latvia Baltic Way TST
P2
Classical inequality
Classical inequality
Source: Latvian TST for Baltic Way 2022 P2
October 25, 2022
inequalities
Problem Statement
Prove that for positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfying
a
b
c
=
1
abc=1
ab
c
=
1
the following inequality holds:
a
b
+
b
c
+
c
a
≥
a
2
+
1
2
a
+
b
2
+
1
2
b
+
c
2
+
1
2
c
.
\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \ge \frac{a^2+1}{2a}+\frac{b^2+1}{2b}+\frac{c^2+1}{2c}.
b
a
+
c
b
+
a
c
≥
2
a
a
2
+
1
+
2
b
b
2
+
1
+
2
c
c
2
+
1
.
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