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2006 Flanders Math Olympiad
1
Trigonometry: 3 roots are given
Trigonometry: 3 roots are given
Source: Flanders MO 2006 Q1
April 19, 2006
trigonometry
algebra
polynomial
Vieta
complex numbers
Problem Statement
(a) Solve for
θ
∈
R
\theta\in\mathbb{R}
θ
∈
R
:
cos
(
4
θ
)
=
cos
(
3
θ
)
\cos(4\theta) = \cos(3\theta)
cos
(
4
θ
)
=
cos
(
3
θ
)
(b)
cos
(
2
π
7
)
\cos\left(\frac{2\pi}{7}\right)
cos
(
7
2
π
)
,
cos
(
4
π
7
)
\cos\left(\frac{4\pi}{7}\right)
cos
(
7
4
π
)
and
cos
(
6
π
7
)
\cos\left(\frac{6\pi}{7}\right)
cos
(
7
6
π
)
are the roots of an equation of the form
a
x
3
+
b
x
2
+
c
x
+
d
=
0
ax^3+bx^2+cx+d = 0
a
x
3
+
b
x
2
+
c
x
+
d
=
0
where
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are integers. Determine
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
.
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