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Baltic Way
2018 Baltic Way
3
Hard cyclic inequality with square roots
Hard cyclic inequality with square roots
Source: Baltic Way 2018, Problem 3
November 6, 2018
inequalities
Problem Statement
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive real numbers such that
a
b
c
d
=
1
abcd=1
ab
c
d
=
1
. Prove the inequality
1
a
+
2
b
+
3
c
+
10
+
1
b
+
2
c
+
3
d
+
10
+
1
c
+
2
d
+
3
a
+
10
+
1
d
+
2
a
+
3
b
+
10
≤
1.
\frac{1}{\sqrt{a+2b+3c+10}}+\frac{1}{\sqrt{b+2c+3d+10}}+\frac{1}{\sqrt{c+2d+3a+10}}+\frac{1}{\sqrt{d+2a+3b+10}} \le 1.
a
+
2
b
+
3
c
+
10
1
+
b
+
2
c
+
3
d
+
10
1
+
c
+
2
d
+
3
a
+
10
1
+
d
+
2
a
+
3
b
+
10
1
≤
1.
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