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Greece Contests
Greece National Olympiad
1998 Greece National Olympiad
4
4
Part of
1998 Greece National Olympiad
Problems
(1)
Prove g(k-1)=g(k)=g(k+1) for non-decreasing g is impossible
Source: Greece MO 1998
5/27/2011
Let a function
g
:
N
0
→
N
0
g:\mathbb{N}_0\to\mathbb{N}_0
g
:
N
0
→
N
0
satisfy
g
(
0
)
=
0
g(0)=0
g
(
0
)
=
0
and
g
(
n
)
=
n
−
g
(
g
(
n
−
1
)
)
g(n)=n-g(g(n-1))
g
(
n
)
=
n
−
g
(
g
(
n
−
1
))
for all
n
≥
1
n\ge 1
n
≥
1
. Prove that:a)
g
(
k
)
≥
g
(
k
−
1
)
g(k)\ge g(k-1)
g
(
k
)
≥
g
(
k
−
1
)
for any positive integer
k
k
k
. b) There is no
k
k
k
such that
g
(
k
−
1
)
=
g
(
k
)
=
g
(
k
+
1
)
g(k-1)=g(k)=g(k+1)
g
(
k
−
1
)
=
g
(
k
)
=
g
(
k
+
1
)
.
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