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IMO Longlists
1985 IMO Longlists
65
Prove the identity for functions
Prove the identity for functions
Source:
September 14, 2010
function
ceiling function
floor function
algebra unsolved
algebra
Problem Statement
Define the functions
f
,
F
:
N
→
N
f, F : \mathbb N \to \mathbb N
f
,
F
:
N
→
N
, by
f
(
n
)
=
[
3
−
5
2
n
]
,
F
(
k
)
=
min
{
n
∈
N
∣
f
k
(
n
)
>
0
}
,
f(n)=\left[ \frac{3-\sqrt 5}{2} n \right] , F(k) =\min \{n \in \mathbb N|f^k(n) > 0 \},
f
(
n
)
=
[
2
3
−
5
n
]
,
F
(
k
)
=
min
{
n
∈
N
∣
f
k
(
n
)
>
0
}
,
where
f
k
=
f
∘
⋯
∘
f
f^k = f \circ \cdots \circ f
f
k
=
f
∘
⋯
∘
f
is
f
f
f
iterated
n
n
n
times. Prove that
F
(
k
+
2
)
=
3
F
(
k
+
1
)
−
F
(
k
)
F(k + 2) = 3F(k + 1) - F(k)
F
(
k
+
2
)
=
3
F
(
k
+
1
)
−
F
(
k
)
for all
k
∈
N
.
k \in \mathbb N.
k
∈
N
.
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