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18
2016-2017 Fall OMO Problem 18
2016-2017 Fall OMO Problem 18
Source:
November 16, 2016
Online Math Open
Problem Statement
Find the smallest positive integer
k
k
k
such that there exist positive integers
M
,
O
>
1
M,O>1
M
,
O
>
1
satisfying
(
O
⋅
M
⋅
O
)
k
=
(
O
⋅
M
)
⋅
(
N
⋅
O
⋅
M
)
⋅
(
N
⋅
O
⋅
M
)
⋅
…
⋅
(
N
⋅
O
⋅
M
)
⏟
2016
(
N
⋅
O
⋅
M
)
s
,
(O\cdot M\cdot O)^k=(O\cdot M)\cdot \underbrace{(N\cdot O\cdot M)\cdot (N\cdot O\cdot M)\cdot \ldots \cdot (N\cdot O\cdot M)}_{2016\ (N\cdot O\cdot M)\text{s}},
(
O
⋅
M
⋅
O
)
k
=
(
O
⋅
M
)
⋅
2016
(
N
⋅
O
⋅
M
)
s
(
N
⋅
O
⋅
M
)
⋅
(
N
⋅
O
⋅
M
)
⋅
…
⋅
(
N
⋅
O
⋅
M
)
,
where
N
=
O
M
N=O^M
N
=
O
M
.Proposed by James Lin and Yannick Yao
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