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Ireland Contests
Ireland National Math Olympiad
1993 Irish Math Olympiad
4
Irish 1993 paper 2 #4
Irish 1993 paper 2 #4
Source:
September 8, 2014
integration
trigonometry
Problem Statement
Let
x
x
x
be a real number with
0
<
x
<
π
0<x<\pi
0
<
x
<
π
.Prove that, for all natural number
n
n
n
,
s
i
n
x
+
s
i
n
3
x
3
+
s
i
n
5
x
5
+
⋯
+
s
i
n
(
2
n
−
1
)
x
2
n
−
1
>
0.
sinx+\frac{sin3x}{3}+\frac{sin5x}{5}+\cdots+\frac{sin(2n-1)x}{2n-1}>0.
s
in
x
+
3
s
in
3
x
+
5
s
in
5
x
+
⋯
+
2
n
−
1
s
in
(
2
n
−
1
)
x
>
0.
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