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District Olympiad
2001 District Olympiad
4
Romania District Olympiad 2001 - Grade IX
Romania District Olympiad 2001 - Grade IX
Source:
March 12, 2011
function
induction
algebra proposed
algebra
Problem Statement
Consider a function
f
:
Z
→
Z
f:\mathbb{Z}\to \mathbb{Z}
f
:
Z
→
Z
such that:
f
(
m
2
+
f
(
n
)
)
=
f
2
(
m
)
+
n
,
∀
m
,
n
∈
Z
f(m^2+f(n))=f^2(m)+n,\ \forall m,n\in \mathbb{Z}
f
(
m
2
+
f
(
n
))
=
f
2
(
m
)
+
n
,
∀
m
,
n
∈
Z
Prove that:a)
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
; b)
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
; c)
f
(
n
)
=
n
,
∀
n
∈
Z
f(n)=n,\ \forall n\in \mathbb{Z}
f
(
n
)
=
n
,
∀
n
∈
Z
Lucian Dragomir
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