MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2021 Turkey Team Selection Test
8
8
Part of
2021 Turkey Team Selection Test
Problems
(1)
Is this function periodic?
Source: Turkey TST 2021 D-3, P-2
5/21/2021
Let
c
c
c
be a real number. For all
x
x
x
and
y
y
y
real numbers we have,
f
(
x
−
f
(
y
)
)
=
f
(
x
−
y
)
+
c
(
f
(
x
)
−
f
(
y
)
)
f(x-f(y))=f(x-y)+c(f(x)-f(y))
f
(
x
−
f
(
y
))
=
f
(
x
−
y
)
+
c
(
f
(
x
)
−
f
(
y
))
and
f
(
x
)
f(x)
f
(
x
)
is not constant.
a
)
a)
a
)
Find all possible values of
c
c
c
.
b
)
b)
b
)
Can
f
f
f
be periodic?
algebra
functional equation
function