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International Contests
Middle European Mathematical Olympiad
2022 Middle European Mathematical Olympiad
1
Two sum-product sequences
Two sum-product sequences
Source: MEMO 2022 T1
September 2, 2022
algebra
Problem Statement
Given a pair
(
a
0
,
b
0
)
(a_0, b_0)
(
a
0
,
b
0
)
of real numbers, we define two sequences
a
0
,
a
1
,
a
2
,
.
.
.
a_0, a_1, a_2,...
a
0
,
a
1
,
a
2
,
...
and
b
0
,
b
1
,
b
2
,
.
.
.
b_0, b_1, b_2, ...
b
0
,
b
1
,
b
2
,
...
of real numbers by
a
n
+
1
=
a
n
+
b
n
a_{n+1}= a_n + b_n
a
n
+
1
=
a
n
+
b
n
and
b
n
+
1
=
a
n
b
n
b_{n+1}=a_nb_n
b
n
+
1
=
a
n
b
n
for all
n
=
0
,
1
,
2
,
.
.
.
n = 0, 1, 2,...
n
=
0
,
1
,
2
,
...
. Find all pairs
(
a
0
,
b
0
)
(a_0, b_0)
(
a
0
,
b
0
)
of real numbers such that
a
2022
=
a
0
a_{2022}= a_0
a
2022
=
a
0
and
b
2022
=
b
0
b_{2022}= b_0
b
2022
=
b
0
.
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