MathDB
Problems
Contests
National and Regional Contests
India Contests
Postal Coaching
2010 Postal Coaching
3
An interesting functional equation
An interesting functional equation
Source:
October 21, 2010
function
algebra unsolved
algebra
Problem Statement
Find all functions
f
:
Z
→
Z
f: \mathbb{Z} \rightarrow \mathbb{Z}
f
:
Z
→
Z
such that
1
f
(
1
)
=
1
\boxed{1} \ f(1) = 1
1
f
(
1
)
=
1
2
f
(
m
+
n
)
(
f
(
m
)
−
f
(
n
)
)
=
f
(
m
−
n
)
(
f
(
m
)
+
f
(
n
)
)
∀
m
,
n
∈
Z
\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}
2
f
(
m
+
n
)
(
f
(
m
)
−
f
(
n
))
=
f
(
m
−
n
)
(
f
(
m
)
+
f
(
n
))
∀
m
,
n
∈
Z
Back to Problems
View on AoPS