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Contests
National and Regional Contests
China Contests
China Team Selection Test
1998 China Team Selection Test
3
Inequality involving sines and cosines
Inequality involving sines and cosines
Source: China TST 1998, problem 3
May 22, 2005
inequalities
trigonometry
inequalities unsolved
Problem Statement
For a fixed
θ
∈
[
0
,
π
2
]
\theta \in \lbrack 0, \frac{\pi}{2} \rbrack
θ
∈
[
0
,
2
π
]
, find the smallest
a
∈
R
+
a \in \mathbb{R}^{+}
a
∈
R
+
which satisfies the following conditions: I.
a
cos
θ
+
a
sin
θ
>
1
\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} > 1
c
o
s
θ
a
+
s
i
n
θ
a
>
1
. II. There exists
x
∈
[
1
−
a
sin
θ
,
a
cos
θ
]
x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta}, \frac{\sqrt a}{\cos \theta} \rbrack
x
∈
[
1
−
s
i
n
θ
a
,
c
o
s
θ
a
]
such that
[
(
1
−
x
)
sin
θ
−
a
−
x
2
cos
2
θ
]
2
+
[
x
cos
θ
−
a
−
(
1
−
x
)
2
sin
2
θ
]
2
≤
a
\lbrack (1 - x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} + \lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta} \rbrack^{2} \leq a
[(
1
−
x
)
sin
θ
−
a
−
x
2
cos
2
θ
]
2
+
[
x
cos
θ
−
a
−
(
1
−
x
)
2
sin
2
θ
]
2
≤
a
.
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