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Problems
Contests
International Contests
Austrian-Polish
1990 Austrian-Polish Competition
6
6
Part of
1990 Austrian-Polish Competition
Problems
(1)
a_1 = a_3 when a_i+1=p(a_1), integer polynomila p(x)
Source: Austrian - Polish 1990 APMC
5/7/2020
p
(
x
)
p(x)
p
(
x
)
is a polynomial with integer coefficients. The sequence of integers
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ... , a_n
a
1
,
a
2
,
...
,
a
n
(where
n
>
2
n > 2
n
>
2
) satisfies
a
2
=
p
(
a
1
)
,
a
3
=
p
(
a
2
)
,
.
.
.
,
a
n
=
p
(
a
n
−
1
)
,
a
1
=
p
(
a
n
)
a_2 = p(a_1), a_3 = p(a_2), ... , a_n = p(a_{n-1}), a_1 = p(a_n)
a
2
=
p
(
a
1
)
,
a
3
=
p
(
a
2
)
,
...
,
a
n
=
p
(
a
n
−
1
)
,
a
1
=
p
(
a
n
)
. Show that
a
1
=
a
3
a_1 = a_3
a
1
=
a
3
.
algebra
polynomial
Integer Polynomial