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2015 Romania National Olympiad
4
Asymmetrical acyclical inequality
Asymmetrical acyclical inequality
Source: Romania NMO 2015,9th grade,Problem 4
April 25, 2015
inequalities
algebra
Problem Statement
Let
a
,
b
,
c
,
d
≥
0
a,b,c,d \ge 0
a
,
b
,
c
,
d
≥
0
real numbers so that
a
+
b
+
c
+
d
=
1
a+b+c+d=1
a
+
b
+
c
+
d
=
1
.Prove that
a
+
(
b
−
c
)
2
6
+
(
c
−
d
)
2
6
+
(
d
−
b
)
2
6
+
b
+
c
+
d
≤
2.
\sqrt{a+\frac{(b-c)^2}{6}+\frac{(c-d)^2}{6}+\frac{(d-b)^2}{6}} +\sqrt{b}+\sqrt{c}+\sqrt{d} \le 2.
a
+
6
(
b
−
c
)
2
+
6
(
c
−
d
)
2
+
6
(
d
−
b
)
2
+
b
+
c
+
d
≤
2.
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