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2002 Polish MO Finals
3
sequence and coprime elements
sequence and coprime elements
Source:
December 9, 2005
induction
number theory proposed
number theory
strong induction
Problem Statement
k
k
k
is a positive integer. The sequence
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
is defined by
a
1
=
k
+
1
a_1 = k+1
a
1
=
k
+
1
,
a
n
+
1
=
a
n
2
−
k
a
n
+
k
a_{n+1} = a_n ^2 - ka_n + k
a
n
+
1
=
a
n
2
−
k
a
n
+
k
. Show that
a
m
a_m
a
m
and
a
n
a_n
a
n
are coprime (for
m
≠
n
m \not = n
m
=
n
).
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