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National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1983 Vietnam National Olympiad
2
Two trigonometric inequalities.
Two trigonometric inequalities.
Source: Vietnam MO 1983 P2
March 18, 2011
trigonometry
inequalities unsolved
inequalities
Problem Statement
(
a
)
(a)
(
a
)
Prove that
2
(
sin
t
+
cos
t
)
≥
2
sin
2
t
4
\sqrt{2}(\sin t + \cos t) \ge 2\sqrt[4]{\sin 2t}
2
(
sin
t
+
cos
t
)
≥
2
4
sin
2
t
for
0
≤
t
≤
π
2
.
0 \le t \le\frac{\pi}{2}.
0
≤
t
≤
2
π
.
(
b
)
(b)
(
b
)
Find all
y
,
0
<
y
<
π
y, 0 < y < \pi
y
,
0
<
y
<
π
, such that
1
+
2
cot
2
y
cot
y
≥
tan
2
y
tan
y
1 +\frac{2 \cot 2y}{\cot y} \ge \frac{\tan 2y}{\tan y}
1
+
c
o
t
y
2
c
o
t
2
y
≥
t
a
n
y
t
a
n
2
y
. .
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