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Spain Mathematical Olympiad
2008 Spain Mathematical Olympiad
2
Spanish inequality for 0<a,b<1
Spanish inequality for 0<a,b<1
Source:
February 11, 2011
inequalities
trigonometry
Spain
Problem Statement
Let
a
a
a
and
b
b
b
be two real numbers, with
0
<
a
,
b
<
1
0<a,b<1
0
<
a
,
b
<
1
. Prove that
a
b
2
+
a
2
b
+
(
1
−
a
)
(
1
−
b
)
2
+
(
1
−
a
)
2
(
1
−
b
)
<
2
\sqrt{ab^2+a^2b}+\sqrt{(1-a)(1-b)^2+(1-a)^2(1-b)}<\sqrt{2}
a
b
2
+
a
2
b
+
(
1
−
a
)
(
1
−
b
)
2
+
(
1
−
a
)
2
(
1
−
b
)
<
2
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