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National and Regional Contests
Romania Contests
Romania Team Selection Test
1992 Romania Team Selection Test
7
7
Part of
1992 Romania Team Selection Test
Problems
(1)
sequence infinitely
Source: Laurentiu Panaitopol, Romania, TST 1992
8/10/2005
Let
(
a
n
)
n
≥
1
(a_{n})_{n\geq 1}
(
a
n
)
n
≥
1
and
(
b
n
)
n
≥
1
(b_{n})_{n\geq 1}
(
b
n
)
n
≥
1
be the sequence of positive integers defined by
a
n
+
1
=
n
a
n
+
1
a_{n+1}=na_{n}+1
a
n
+
1
=
n
a
n
+
1
and
b
n
+
1
=
n
b
n
−
1
b_{n+1}=nb_{n}-1
b
n
+
1
=
n
b
n
−
1
for
n
≥
1
n\geq 1
n
≥
1
. Show that the two sequence cannot have infinitely many common terms. Laurentiu Panaitopol
algebra proposed
algebra