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National and Regional Contests
Malaysia Contests
JOM Shortlists
JOM 2015 Shortlist
A8
A8
Part of
JOM 2015 Shortlist
Problems
(1)
Sequence
Source: Junior Olympiad of Malaysia Shortlist 2015 A8
7/17/2015
Let
a
1
,
a
2
,
⋯
,
a
2015
a_1,a_2, \cdots ,a_{2015}
a
1
,
a
2
,
⋯
,
a
2015
be
2015
2015
2015
-tuples of positive integers (not necessary distinct) and let
k
k
k
be a positive integers. Denote
f
(
i
)
=
a
i
+
a
1
a
2
⋯
a
2015
a
i
\displaystyle f(i)=a_i+\frac{a_1a_2 \cdots a_{2015}}{a_i}
f
(
i
)
=
a
i
+
a
i
a
1
a
2
⋯
a
2015
. a) Prove that if
k
=
201
5
2015
k=2015^{2015}
k
=
201
5
2015
, there exist
a
1
,
a
2
,
⋯
,
a
2015
a_1, a_2, \cdots , a_{2015}
a
1
,
a
2
,
⋯
,
a
2015
such that
f
(
i
)
=
k
f(i)= k
f
(
i
)
=
k
for all
1
≤
i
≤
2015
1\le i\le 2015
1
≤
i
≤
2015
.\\ b) Find the maximum
k
0
k_0
k
0
so that for
k
≤
k
0
k\le k_0
k
≤
k
0
, there are no
k
k
k
such that there are at least
2
2
2
different
2015
2015
2015
-tuples which fulfill the above condition.
algebra