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Thailand National Olympiad
2015 Thailand Mathematical Olympiad
1
1
Part of
2015 Thailand Mathematical Olympiad
Problems
(1)
a_{n+1} divides a_n+a_{n+2} when a_n a_{n+2} = a^2_{n+1} + p
Source: Thailand Mathematical Olympiad 2015 p1
8/16/2020
Let
p
p
p
be a prime, and let
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, . . .
a
1
,
a
2
,
a
3
,
...
be a sequence of positive integers so that
a
n
a
n
+
2
=
a
n
+
1
2
+
p
a_na_{n+2} = a^2_{n+1} + p
a
n
a
n
+
2
=
a
n
+
1
2
+
p
for all positive integers
n
n
n
. Show that
a
n
+
1
a_{n+1}
a
n
+
1
divides
a
n
+
a
n
+
2
a_n + a_{n+2}
a
n
+
a
n
+
2
for all positive integers
n
n
n
.
divides
recurrence relation
divisible
number theory