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International Contests
Austrian-Polish
1981 Austrian-Polish Competition
2
2
Part of
1981 Austrian-Polish Competition
Problems
(1)
a_{n+1} = a_n^2 + (a_n - 1)^2 , a_q - ap = a_m - a_k
Source: Austrian Polish 1981 APMC
4/26/2020
The sequence
a
0
,
a
1
,
a
2
,
.
.
.
a_0, a_1, a_2, ...
a
0
,
a
1
,
a
2
,
...
is defined by
a
n
+
1
=
a
n
2
+
(
a
n
−
1
)
2
a_{n+1} = a^2_n + (a_n - 1)^2
a
n
+
1
=
a
n
2
+
(
a
n
−
1
)
2
for
n
≥
0
n \ge 0
n
≥
0
. Find all rational numbers
a
0
a_0
a
0
for which there exist four distinct indices
k
,
m
,
p
,
q
k, m, p, q
k
,
m
,
p
,
q
such that
a
q
−
a
p
=
a
m
−
a
k
a_q - a_p = a_m - a_k
a
q
−
a
p
=
a
m
−
a
k
.
Sequence
recurrence relation
algebra
rational