MathDB
Problems
Contests
National and Regional Contests
Latvia Contests
Latvia BW TST
2019 Latvia Baltic Way TST
1
1
Part of
2019 Latvia Baltic Way TST
Problems
(1)
Not homogenous, messy inequality
Source: Latvian TST for Baltic Way 2019 Problem 1
5/29/2020
Prove that for all positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
with
1
a
+
1
b
+
1
c
=
1
\frac{1}{a}+\frac{1}{b}+\frac{1}{c} =1
a
1
+
b
1
+
c
1
=
1
the following inequality holds:
3
(
a
b
+
b
c
+
c
a
)
+
9
a
+
b
+
c
≤
9
a
b
c
a
+
b
+
c
+
2
(
a
2
+
b
2
+
c
2
)
+
1
3(ab+bc+ca)+\frac{9}{a+b+c} \le \frac{9abc}{a+b+c} + 2(a^2+b^2+c^2)+1
3
(
ab
+
b
c
+
c
a
)
+
a
+
b
+
c
9
≤
a
+
b
+
c
9
ab
c
+
2
(
a
2
+
b
2
+
c
2
)
+
1
inequalities