MathDB
Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2015 Irish Math Olympiad
5
5
Part of
2015 Irish Math Olympiad
Problems
(1)
Bounded sequence is constant
Source:
10/31/2015
Suppose a doubly infinite sequence of real numbers
.
.
.
,
a
−
2
,
a
−
1
,
a
0
,
a
1
,
a
2
,
.
.
.
. . . , a_{-2}, a_{-1}, a_0, a_1, a_2, . . .
...
,
a
−
2
,
a
−
1
,
a
0
,
a
1
,
a
2
,
...
has the property that
a
n
+
3
=
a
n
+
a
n
+
1
+
a
n
+
2
3
,
a_{n+3} =\frac{a_n + a_{n+1} + a_{n+2}}{3},
a
n
+
3
=
3
a
n
+
a
n
+
1
+
a
n
+
2
,
for all integers
n
.
n .
n
.
Show that if this sequence is bounded (i.e., if there exists a number
R
R
R
such that
∣
a
n
∣
≤
R
|a_n| \leq R
∣
a
n
∣
≤
R
for all
n
n
n
), then
a
n
a_n
a
n
has the same value for all
n
.
n.
n
.
algebra