MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canadian Open Math Challenge
2017 Canadian Open Math Challenge
C2
C2
Part of
2017 Canadian Open Math Challenge
Problems
(1)
2017 COMC C2
Source:
10/12/2018
Source: 2017 Canadian Open Math Challenge, Problem C2 —-- A function
f
(
x
)
f(x)
f
(
x
)
is periodic with period
T
>
0
T > 0
T
>
0
if
f
(
x
+
T
)
=
f
(
x
)
f(x + T) = f(x)
f
(
x
+
T
)
=
f
(
x
)
for all
x
x
x
. The smallest such number
T
T
T
is called the least period. For example, the functions
sin
(
x
)
\sin(x)
sin
(
x
)
and
cos
(
x
)
\cos(x)
cos
(
x
)
are periodic with least period
2
π
2\pi
2
π
.
\qquad
(a) Let a function
g
(
x
)
g(x)
g
(
x
)
be periodic with the least period
T
=
π
T = \pi
T
=
π
. Determine the least period of
g
(
x
/
3
)
g(x/3)
g
(
x
/3
)
.
\qquad
(b) Determine the least period of
H
(
x
)
=
s
i
n
(
8
x
)
+
c
o
s
(
4
x
)
H(x) = sin(8x) + cos(4x)
H
(
x
)
=
s
in
(
8
x
)
+
cos
(
4
x
)
\qquad
(c) Determine the least periods of each of
G
(
x
)
=
s
i
n
(
c
o
s
(
x
)
)
G(x) = sin(cos(x))
G
(
x
)
=
s
in
(
cos
(
x
))
and
F
(
x
)
=
c
o
s
(
s
i
n
(
x
)
)
F(x) = cos(sin(x))
F
(
x
)
=
cos
(
s
in
(
x
))
.
Comc
2017 COMC