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National and Regional Contests
PEN Problems
PEN M Problems
1
M 1
M 1
Source:
May 25, 2007
algebra
polynomial
induction
Recursive Sequences
Problem Statement
Let
P
(
x
)
P(x)
P
(
x
)
be a nonzero polynomial with integer coefficients. Let
a
0
=
0
a_{0}=0
a
0
=
0
and for
i
≥
0
i \ge 0
i
≥
0
define
a
i
+
1
=
P
(
a
i
)
a_{i+1}=P(a_{i})
a
i
+
1
=
P
(
a
i
)
. Show that
gcd
(
a
m
,
a
n
)
=
a
gcd
(
m
,
n
)
\gcd ( a_{m}, a_{n})=a_{ \gcd (m, n)}
g
cd
(
a
m
,
a
n
)
=
a
g
c
d
(
m
,
n
)
for all
m
,
n
∈
N
m, n \in \mathbb{N}
m
,
n
∈
N
.
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