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APMO
2020 APMO
2
2
Part of
2020 APMO
Problems
(1)
Sequence satisfying sqrt inequality eventually alternates
Source: APMO 2020 Problem 2
6/9/2020
Show that
r
=
2
r = 2
r
=
2
is the largest real number
r
r
r
which satisfies the following condition:If a sequence
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\ldots
…
of positive integers fulfills the inequalities
a
n
≤
a
n
+
2
≤
a
n
2
+
r
a
n
+
1
a_n \leq a_{n+2} \leq\sqrt{a_n^2+ra_{n+1}}
a
n
≤
a
n
+
2
≤
a
n
2
+
r
a
n
+
1
for every positive integer
n
n
n
, then there exists a positive integer
M
M
M
such that
a
n
+
2
=
a
n
a_{n+2} = a_n
a
n
+
2
=
a
n
for every
n
≥
M
n \geq M
n
≥
M
.
APMO 2020