Problem A. 566. (September 2012)
A. 566. (a) Prove that if n2 and the product of the positive real numbers is 1 then . (b) Show an example for an integer n2 and positive real numbers having product 1 that satisfy .
(5 pont)
Deadline expired on 10 October 2012.
Solution. (a) For k3, apply the AMGM inequality to 1/(k2) and a_{k}/2 with the weights k2 and 2:
 (1) 
We have equality if 1/(k2)=a_{k}/2, so .
Taking the product for ,
 (2) 
The trivial estimate 1+a_{2}^{2}>a_{2}^{2} completes the proof.
(b) We have equality in (2) if for k3; then the constraint enforces .
It is easy to find that for n=14 we have and , so
Hence, a possible example is
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