MathDB
Problems
Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2012 Bosnia Herzegovina Team Selection Test
2
Bosnia and Herzegovina TST 2012 Problem 2
Bosnia and Herzegovina TST 2012 Problem 2
Source:
May 19, 2012
trigonometry
inequalities
inequalities proposed
Problem Statement
Prove for all positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
, such that
a
2
+
b
2
+
c
2
=
1
a^2+b^2+c^2=1
a
2
+
b
2
+
c
2
=
1
:
a
3
b
2
+
c
+
b
3
c
2
+
a
+
c
3
a
2
+
b
≥
3
1
+
3
.
\frac{a^3}{b^2+c}+\frac{b^3}{c^2+a}+\frac{c^3}{a^2+b}\ge \frac{\sqrt{3}}{1+\sqrt{3}}.
b
2
+
c
a
3
+
c
2
+
a
b
3
+
a
2
+
b
c
3
≥
1
+
3
3
.
Back to Problems
View on AoPS