Difference between revisions of "G285 Summer Problem Set Problem 8"
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Latest revision as of 12:50, 24 June 2021
Problem 8
Let , Let
be the twelve roots that satisfies
, find the least possible value of
Solution
The first summation is simply by Vieta's. The second summation is
. The minimum possible value is
, so we need to minimize
. If we do bounding, when
we have
, and when
we have
. The shift implies there is a root
where
such that
. However,
seems very close to
, and
approaches infinity, so there is another root
. Therefore, we have the smallest root
must be
, where
is the fractional part. The answer is