Problem A. 391. (January 2006)
A. 391. Construct a sequence a_{1},a_{2},...,a_{N} of positive reals such that n_{1}a_{n0}+n_{2}a_{n1}+...+n_{k}a_{nk1}>2.7(a_{1}+a_{2}+...+a_{N}) for arbitrary integers 1=n_{0}<n_{1}<...<n_{k}=N.
(5 pont)
Deadline expired on February 15, 2006.
Solution. Let N be sufficiently large (it will be defined later) and set . Then
and for any sequence 1=n_{0}<n_{1}<...<n_{k}=N of indices we have
The minimum of the function is at point , its value is e^{.}log N. Therefore,
If N is chosen such that then e^{.}log N>2,7^{.}(1+log N).
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