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International Contests
Austrian-Polish
1992 Austrian-Polish Competition
6
6
Part of
1992 Austrian-Polish Competition
Problems
(1)
f (92+ x) = f (92- x),f (19 .92+x) = f(19. 92-x),f (1992 + x) = f (1992 -x)
Source: Austrian - Polish 1992 APMC
5/7/2020
A function
f
:
Z
→
Z
f: Z \to Z
f
:
Z
→
Z
has the following properties:
f
(
92
+
x
)
=
f
(
92
−
x
)
f (92 + x) = f (92 - x)
f
(
92
+
x
)
=
f
(
92
−
x
)
f
(
19
⋅
92
+
x
)
=
f
(
19
⋅
92
−
x
)
f (19 \cdot 92 + x) = f (19 \cdot 92 - x)
f
(
19
⋅
92
+
x
)
=
f
(
19
⋅
92
−
x
)
(
19
⋅
92
=
1748
19 \cdot 92 = 1748
19
⋅
92
=
1748
)
f
(
1992
+
x
)
=
f
(
1992
−
x
)
f (1992 + x) = f (1992 - x)
f
(
1992
+
x
)
=
f
(
1992
−
x
)
for all integers
x
x
x
. Can all positive divisors of
92
92
92
occur as values of f?
functional
functional equation
Divisors
algebra
number theory