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Bulgaria Contests
Bulgarian Spring Mathematical Competition
2023 Bulgarian Spring Mathematical Competition
11.1
11.1
Part of
2023 Bulgarian Spring Mathematical Competition
Problems
(1)
Equation with powers of 3 and trig
Source: Bulgarian Spring Tournament 2023 11.1
3/25/2023
Find all real
a
a
a
such that the equation
3
cos
(
2
x
)
+
1
−
(
a
−
5
)
3
cos
2
(
2
x
)
=
7
3^{\cos (2x)+1}-(a-5)3^{\cos^2(2x)}=7
3
c
o
s
(
2
x
)
+
1
−
(
a
−
5
)
3
c
o
s
2
(
2
x
)
=
7
has a real root. This was the statement given at the contest, but there was actually a typo and the intended equation was
3
cos
(
2
x
)
+
1
−
(
a
−
5
)
3
cos
2
(
x
)
=
7
3^{\cos (2x)+1}-(a-5)3^{\cos^2(x)}=7
3
c
o
s
(
2
x
)
+
1
−
(
a
−
5
)
3
c
o
s
2
(
x
)
=
7
, which is much easier.
trigonometry
algebra