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Romanian Masters of Mathematics Collection
2016 Romanian Master of Mathematics
3
3
Part of
2016 Romanian Master of Mathematics
Problems
(1)
Cubic sequence
Source: RMM 2016 Day 1 Problem 3
2/27/2016
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<span class='latex-italic'>cubic sequence</span>
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is a sequence of integers given by
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+
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+
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a_n =n^3 + bn^2 + cn + d
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+
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+
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, where
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b, c
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and
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d
are integer constants and
n
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n
ranges over all integers, including negative integers.
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<span class='latex-bold'>(a)</span>
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Show that there exists a cubic sequence such that the only terms of the sequence which are squares of integers are
a
2015
a_{2015}
a
2015
and
a
2016
a_{2016}
a
2016
.
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<span class='latex-bold'>(b)</span>
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Determine the possible values of
a
2015
⋅
a
2016
a_{2015} \cdot a_{2016}
a
2015
⋅
a
2016
for a cubic sequence satisfying the condition in part
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<span class='latex-bold'>(a)</span>
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.
number theory
RMM