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International Contests
Balkan MO
1985 Balkan MO
2
trigonometry
trigonometry
Source: bmo 1985
April 23, 2007
trigonometry
function
inequalities
quadratics
algebra proposed
algebra
Problem Statement
Let
a
,
b
,
c
,
d
∈
[
−
π
2
,
π
2
]
a,b,c,d \in [-\frac{\pi}{2}, \frac{\pi}{2}]
a
,
b
,
c
,
d
∈
[
−
2
π
,
2
π
]
be real numbers such that
sin
a
+
sin
b
+
sin
c
+
sin
d
=
1
\sin{a}+\sin{b}+\sin{c}+\sin{d}=1
sin
a
+
sin
b
+
sin
c
+
sin
d
=
1
and
cos
2
a
+
cos
2
b
+
cos
2
c
+
cos
2
d
≥
10
3
\cos{2a}+\cos{2b}+\cos{2c}+\cos{2d}\geq \frac{10}{3}
cos
2
a
+
cos
2
b
+
cos
2
c
+
cos
2
d
≥
3
10
. Prove that
a
,
b
,
c
,
d
∈
[
0
,
π
6
]
a,b,c,d \in [0, \frac{\pi}{6}]
a
,
b
,
c
,
d
∈
[
0
,
6
π
]
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