Problem A. 391. (January 2006)
A. 391. Construct a sequence a1,a2,...,aN of positive reals such that n1an0+n2an1+...+nkank-1>2.7(a1+a2+...+aN) for arbitrary integers 1=n0<n1<...<nk=N.
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=n0<n1<...<nk=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).