Problem A. 478. (March 2009)
A. 478. If a1,a2,...,an are positive numbers with a1+a2+...+an=1, prove .
(5 pont)
Deadline expired on April 15, 2009.
Solution. Let
Apply Hölder's innequality on the sequences (ai1/3) and (aiai+1)1/3 (i=1,2,...,n-1), with the expononts p=3 and q=3/2:
From the estimates and
we obtain
Based on the solution of Nagy János
Statistics:
3 students sent a solution. 5 points: Nagy 235 János, Nagy 314 Dániel, Tomon István.
Problems in Mathematics of KöMaL, March 2009