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Cono Sur Olympiad
2017 Cono Sur Olympiad
5
5
Part of
2017 Cono Sur Olympiad
Problems
(1)
Three sequences
Source: Cono Sur Olympiad 2017, problem 5
8/21/2017
Let
a
a
a
,
b
b
b
and
c
c
c
positive integers. Three sequences are defined as follows:[*]
a
1
=
a
a_1=a
a
1
=
a
,
b
1
=
b
b_1=b
b
1
=
b
,
c
1
=
c
c_1=c
c
1
=
c
[/*] [*]
a
n
+
1
=
⌊
a
n
b
n
⌋
a_{n+1}=\lfloor{\sqrt{a_nb_n}}\rfloor
a
n
+
1
=
⌊
a
n
b
n
⌋
,
b
n
+
1
=
⌊
b
n
c
n
⌋
\:b_{n+1}=\lfloor{\sqrt{b_nc_n}}\rfloor
b
n
+
1
=
⌊
b
n
c
n
⌋
,
c
n
+
1
=
⌊
c
n
a
n
⌋
\:c_{n+1}=\lfloor{\sqrt{c_na_n}}\rfloor
c
n
+
1
=
⌊
c
n
a
n
⌋
for
n
≥
1
n \ge 1
n
≥
1
[/*] [list = a] [*]Prove that for any
a
a
a
,
b
b
b
,
c
c
c
, there exists a positive integer
N
N
N
such that
a
N
=
b
N
=
c
N
a_N=b_N=c_N
a
N
=
b
N
=
c
N
.[/*] [*]Find the smallest
N
N
N
such that
a
N
=
b
N
=
c
N
a_N=b_N=c_N
a
N
=
b
N
=
c
N
for some choice of
a
a
a
,
b
b
b
,
c
c
c
such that
a
≥
2
a \ge 2
a
≥
2
y
b
+
c
=
2
a
−
1
b+c=2a-1
b
+
c
=
2
a
−
1
.[/*]
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