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Problems
Contests
National and Regional Contests
PEN Problems
PEN M Problems
3
3
Part of
PEN M Problems
Problems
(1)
M 3
Source:
5/25/2007
Let
f
(
n
)
=
n
+
⌊
n
⌋
f(n)=n+\lfloor \sqrt{n}\rfloor
f
(
n
)
=
n
+
⌊
n
⌋
. Prove that, for every positive integer
m
m
m
, the sequence
m
,
f
(
m
)
,
f
(
f
(
m
)
)
,
f
(
f
(
f
(
m
)
)
)
,
⋯
m, f(m), f(f(m)), f(f(f(m))), \cdots
m
,
f
(
m
)
,
f
(
f
(
m
))
,
f
(
f
(
f
(
m
)))
,
⋯
contains at least one square of an integer.
floor function
Recursive Sequences