MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2016 Bulgaria National Olympiad
Problem 3
Another nice Inequality
Another nice Inequality
Source: Bulgaria 2016,P 3
July 16, 2016
inequalities
algebra
inequalities proposed
Problem Statement
For
a
,
b
,
c
,
d
>
0
a,b,c,d>0
a
,
b
,
c
,
d
>
0
prove that
a
+
a
b
+
a
b
c
3
+
a
b
c
d
4
4
≤
a
.
a
+
b
2
.
a
+
b
+
c
3
.
a
+
b
+
c
+
d
4
4
\frac {a+\sqrt{ab}+\sqrt[3]{abc}+\sqrt[4]{abcd}}{4} \leq \sqrt[4]{a.\frac{a+b}{2}.\frac{a+b+c}{3}.\frac{a+b+c+d}{4}}
4
a
+
ab
+
3
ab
c
+
4
ab
c
d
≤
4
a
.
2
a
+
b
.
3
a
+
b
+
c
.
4
a
+
b
+
c
+
d
Back to Problems
View on AoPS