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National High School Mathematics League
1996 National High School Mathematics League
2
Trigonometry
Trigonometry
Source: 1996 National High School Mathematics League, Exam Two, Problem 2
March 4, 2020
trigonometry
Problem Statement
Find the range value of
a
a
a
, satisfyin that
∀
x
∈
R
,
θ
∈
[
0
,
π
2
]
\forall x\in\mathbb{R},\theta\in\left[0,\frac{\pi}{2}\right]
∀
x
∈
R
,
θ
∈
[
0
,
2
π
]
,
(
x
+
3
+
2
sin
θ
cos
θ
)
2
+
(
x
+
a
sin
θ
+
a
cos
θ
)
2
≥
1
8
.
(x+3+2\sin\theta\cos\theta)^2+(x+a\sin\theta+a\cos\theta)^2\geq\frac{1}{8}.
(
x
+
3
+
2
sin
θ
cos
θ
)
2
+
(
x
+
a
sin
θ
+
a
cos
θ
)
2
≥
8
1
.
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