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IMO Shortlist
2016 IMO Shortlist
A1
A1
Part of
2016 IMO Shortlist
Problems
(1)
3-variable inequality with min(ab,bc,ca)>=1
Source: 2016 IMO Shortlist A1
7/19/2017
Let
a
a
a
,
b
b
b
,
c
c
c
be positive real numbers such that
min
(
a
b
,
b
c
,
c
a
)
≥
1
\min(ab,bc,ca) \ge 1
min
(
ab
,
b
c
,
c
a
)
≥
1
. Prove that
(
a
2
+
1
)
(
b
2
+
1
)
(
c
2
+
1
)
3
≤
(
a
+
b
+
c
3
)
2
+
1.
\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.
3
(
a
2
+
1
)
(
b
2
+
1
)
(
c
2
+
1
)
≤
(
3
a
+
b
+
c
)
2
+
1.
Proposed by Tigran Margaryan, Armenia
IMO Shortlist
inequalities
three variable inequality
Hi