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The Philippines Contests
Philippine MO
2012 Philippine MO
3
14th Philippine Mathematical Olympiad 2011-National Stage #3
14th Philippine Mathematical Olympiad 2011-National Stage #3
Source:
October 13, 2014
trigonometry
inequalities
inequalities unsolved
Problem Statement
If
a
b
>
0
ab>0
ab
>
0
and
0
<
x
<
π
2
\displaystyle 0<x<\frac{\pi}{2}
0
<
x
<
2
π
, prove that
(
1
+
a
2
sin
x
)
(
1
+
b
2
cos
x
)
≥
(
1
+
2
a
b
)
2
sin
2
x
2
.
\left ( 1+\frac{a^2}{\sin x} \right ) \left ( 1+\frac{b^2}{\cos x} \right ) \geq \frac{(1+\sqrt{2}ab)^2 \sin 2x}{2}.
(
1
+
sin
x
a
2
)
(
1
+
cos
x
b
2
)
≥
2
(
1
+
2
ab
)
2
sin
2
x
.
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