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Cono Sur Olympiad
2017 Cono Sur Olympiad
6
6
Part of
2017 Cono Sur Olympiad
Problems
(1)
Sequence with nested square roots
Source: Cono Sur Olympiad 2017, problem 6
8/21/2017
The infinite sequence
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3,\ldots
a
1
,
a
2
,
a
3
,
…
of positive integers is defined as follows:
a
1
=
1
a_1=1
a
1
=
1
, and for each
n
≥
2
n \ge 2
n
≥
2
,
a
n
a_n
a
n
is the smallest positive integer, distinct from
a
1
,
a
2
,
…
,
a
n
−
1
a_1,a_2, \ldots , a_{n-1}
a
1
,
a
2
,
…
,
a
n
−
1
such that:
a
n
+
a
n
−
1
+
…
+
a
2
+
a
1
\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}}
a
n
+
a
n
−
1
+
…
+
a
2
+
a
1
is an integer. Prove that all positive integers appear on the sequence
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3,\ldots
a
1
,
a
2
,
a
3
,
…
cono sur
number theory