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National and Regional Contests
Iran Contests
Iran Team Selection Test
2011 Iran Team Selection Test
10
10
Part of
2011 Iran Team Selection Test
Problems
(1)
The least value of k for which the inequality holds
Source: Iran TST 2011 - Day 4 - Problem 1
5/14/2011
Find the least value of
k
k
k
such that for all
a
,
b
,
c
,
d
∈
R
a,b,c,d \in \mathbb{R}
a
,
b
,
c
,
d
∈
R
the inequality
(
a
2
+
1
)
(
b
2
+
1
)
(
c
2
+
1
)
+
(
b
2
+
1
)
(
c
2
+
1
)
(
d
2
+
1
)
+
(
c
2
+
1
)
(
d
2
+
1
)
(
a
2
+
1
)
+
(
d
2
+
1
)
(
a
2
+
1
)
(
b
2
+
1
)
≥
2
(
a
b
+
b
c
+
c
d
+
d
a
+
a
c
+
b
d
)
−
k
\begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\ \ \\ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}
(
a
2
+
1
)
(
b
2
+
1
)
(
c
2
+
1
)
+
(
b
2
+
1
)
(
c
2
+
1
)
(
d
2
+
1
)
+
(
c
2
+
1
)
(
d
2
+
1
)
(
a
2
+
1
)
+
(
d
2
+
1
)
(
a
2
+
1
)
(
b
2
+
1
)
≥
2
(
ab
+
b
c
+
c
d
+
d
a
+
a
c
+
b
d
)
−
k
holds.
inequalities
calculus
inequalities proposed