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Inequality on trigonometric circle the arcs - ISL 1975
Inequality on trigonometric circle the arcs - ISL 1975
Source:
September 21, 2010
trigonometry
Trigonometric inequality
Trigonometric Identities
Inequality
optimization
IMO Shortlist
Problem Statement
Consider on the first quadrant of the trigonometric circle the arcs
A
M
1
=
x
1
,
A
M
2
=
x
2
,
A
M
3
=
x
3
,
…
,
A
M
v
=
x
v
AM_1 = x_1,AM_2 = x_2,AM_3 = x_3, \ldots , AM_v = x_v
A
M
1
=
x
1
,
A
M
2
=
x
2
,
A
M
3
=
x
3
,
…
,
A
M
v
=
x
v
, such that
x
1
<
x
2
<
x
3
<
⋯
<
x
v
x_1 < x_2 < x_3 < \cdots < x_v
x
1
<
x
2
<
x
3
<
⋯
<
x
v
. Prove that
∑
i
=
0
v
−
1
sin
2
x
i
−
∑
i
=
0
v
−
1
sin
(
x
i
−
x
i
+
1
)
<
π
2
+
∑
i
=
0
v
−
1
sin
(
x
i
+
x
i
+
1
)
\sum_{i=0}^{v-1} \sin 2x_i - \sum_{i=0}^{v-1} \sin (x_i- x_{i+1}) < \frac{\pi}{2} + \sum_{i=0}^{v-1} \sin (x_i + x_{i+1})
i
=
0
∑
v
−
1
sin
2
x
i
−
i
=
0
∑
v
−
1
sin
(
x
i
−
x
i
+
1
)
<
2
π
+
i
=
0
∑
v
−
1
sin
(
x
i
+
x
i
+
1
)
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