MathDB
Problems
Contests
National and Regional Contests
Latvia Contests
Latvia BW TST
2022 Latvia Baltic Way TST
P2
P2
Part of
2022 Latvia Baltic Way TST
Problems
(1)
Classical inequality
Source: Latvian TST for Baltic Way 2022 P2
10/25/2022
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
.
inequalities