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1966 Bulgaria National Olympiad
Problem 2
inequality, four variables in R+ (Bulgaria 1966 P2)
inequality, four variables in R+ (Bulgaria 1966 P2)
Source:
June 23, 2021
inequalities
Problem Statement
Prove that for every four positive numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
the following inequality is true:
a
2
+
b
2
+
c
2
+
d
2
4
≥
a
b
c
+
a
b
d
+
a
c
d
+
b
c
d
4
3
.
\sqrt{\frac{a^2+b^2+c^2+d^2}4}\ge\sqrt[3]{\frac{abc+abd+acd+bcd}4}.
4
a
2
+
b
2
+
c
2
+
d
2
≥
3
4
ab
c
+
ab
d
+
a
c
d
+
b
c
d
.
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