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National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
1999 Ukraine Team Selection Test
4
sum sin(n+1)x/sin nx < 2cos x/sin^2 x
sum sin(n+1)x/sin nx < 2cos x/sin^2 x
Source: Ukrainian TST 1999 p4
February 13, 2020
inequalities
trigonometry
algebra
Problem Statement
If
n
∈
N
n \in N
n
∈
N
and
0
<
x
<
π
2
n
0 < x <\frac{\pi}{2n}
0
<
x
<
2
n
π
, prove the inequality
sin
2
x
sin
x
+
sin
3
x
sin
2
x
+
.
.
.
+
sin
(
n
+
1
)
x
sin
n
x
<
2
cos
x
sin
2
x
\frac{\sin 2x}{\sin x}+\frac{\sin 3x}{\sin 2x} +...+\frac{\sin (n+1)x}{\sin nx} < 2\frac{\cos x}{\sin^2 x}
s
i
n
x
s
i
n
2
x
+
s
i
n
2
x
s
i
n
3
x
+
...
+
s
i
n
n
x
s
i
n
(
n
+
1
)
x
<
2
s
i
n
2
x
c
o
s
x
. .
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