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2001 IMC
6
IMC 2001 Problem 12
IMC 2001 Problem 12
Source: IMC 2001 Day 2 Problem 6
October 30, 2020
inequalities
derivative
Problem Statement
For each positive integer
n
n
n
, let
f
n
(
ϑ
)
=
sin
(
ϑ
)
⋅
sin
(
2
ϑ
)
⋅
sin
(
4
ϑ
)
⋯
sin
(
2
n
ϑ
)
f_{n}(\vartheta)=\sin(\vartheta)\cdot \sin(2\vartheta) \cdot \sin(4\vartheta)\cdots \sin(2^{n}\vartheta)
f
n
(
ϑ
)
=
sin
(
ϑ
)
⋅
sin
(
2
ϑ
)
⋅
sin
(
4
ϑ
)
⋯
sin
(
2
n
ϑ
)
. For each real
ϑ
\vartheta
ϑ
and all
n
n
n
, prove that
∣
f
n
(
ϑ
)
∣
≤
2
3
∣
f
n
(
π
3
)
∣
|f_{n}(\vartheta)| \leq \frac{2}{\sqrt{3}}|f_{n}(\frac{\pi}{3})|
∣
f
n
(
ϑ
)
∣
≤
3
2
∣
f
n
(
3
π
)
∣
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