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2011 Ukraine Team Selection Test
1
sum \sin (<A _i MA_i)> sin 2 \pi/n + (n-2) sin \pi /n
sum \sin (<A _i MA_i)> sin 2 \pi/n + (n-2) sin \pi /n
Source: Ukraine TST 2011 p1
May 7, 2020
inequalities
trigonometry
Problem Statement
Given a right
n
n
n
-angle
A
1
A
2
…
A
n
{{A} _ {1}} {{A} _ {2}} \ldots {{A} _ {n}}
A
1
A
2
…
A
n
,
n
≥
4
n \ge 4
n
≥
4
, and a point
M
M
M
inside it. Prove the inequality
sin
(
∠
A
1
M
A
2
)
+
sin
(
∠
A
2
M
A
3
)
+
…
+
sin
(
∠
A
n
M
A
1
)
>
sin
2
π
n
+
(
n
−
2
)
s
i
n
π
n
\sin (\angle {{A} _ {1}} M {{A} _ {2}}) + \sin (\angle {{A} _ {2}} M {{A} _ {3}} ) + \ldots + \sin (\angle {{A} _ {n}} M {{A} _ {1}})> \sin \frac{2 \pi}{n} + (n-2) sin \frac{\pi}{n}
sin
(
∠
A
1
M
A
2
)
+
sin
(
∠
A
2
M
A
3
)
+
…
+
sin
(
∠
A
n
M
A
1
)
>
sin
n
2
π
+
(
n
−
2
)
s
in
n
π
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