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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.

(5 points)

Deadline expired on 15 February 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).

Statistics on problem A. 391.
 4 students sent a solution. 5 points: Paulin Roland, Tomon István. 0 point: 2 students.

• Problems in Mathematics of KöMaL, January 2006

•  Támogatóink: Morgan Stanley