MathDB
Problems
Contests
International Contests
IMO Shortlist
1983 IMO Shortlist
7
7
Part of
1983 IMO Shortlist
Problems
(1)
Each term of the sequence is an integer
Source:
9/9/2010
Let
a
a
a
be a positive integer and let
{
a
n
}
\{a_n\}
{
a
n
}
be defined by
a
0
=
0
a_0 = 0
a
0
=
0
and
a
n
+
1
=
(
a
n
+
1
)
a
+
(
a
+
1
)
a
n
+
2
a
(
a
+
1
)
a
n
(
a
n
+
1
)
(
n
=
1
,
2
,
…
)
.
a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).
a
n
+
1
=
(
a
n
+
1
)
a
+
(
a
+
1
)
a
n
+
2
a
(
a
+
1
)
a
n
(
a
n
+
1
)
(
n
=
1
,
2
,
…
)
.
Show that for each positive integer
n
n
n
,
a
n
a_n
a
n
is a positive integer.
number theory
Sequence
recurrence relation
IMO Shortlist
algebra