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National and Regional Contests
PEN Problems
PEN O Problems
8
O 8
O 8
Source:
May 25, 2007
induction
Problem Statement
Let
a
a
a
and
b
b
b
be positive integers greater than
2
2
2
. Prove that there exists a positive integer
k
k
k
and a finite sequence
n
1
n_1
n
1
,
⋯
\cdots
⋯
,
n
k
n_k
n
k
of positive integers such that
n
1
=
a
n_1 =a
n
1
=
a
,
n
k
=
b
n_k =b
n
k
=
b
, and
n
i
n
i
+
1
n_i n_{i+1}
n
i
n
i
+
1
is divisible by
n
i
+
n
i
+
1
n_{i}+n_{i+1}
n
i
+
n
i
+
1
for each
i
i
i
(
1
≤
i
≤
k
)
(1 \le i \le k)
(
1
≤
i
≤
k
)
.
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