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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 points)

Deadline expired on 10 October 2012.

Solution. (a) For k3, apply the AM-GM inequality to 1/(k-2) and ak/2 with the weights k-2 and 2:

 (1)

We have equality if 1/(k-2)=ak/2, so .

Taking the product for ,

 (2)

The trivial estimate 1+a22>a22 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

Statistics on problem A. 566.
 7 students sent a solution. 5 points: Herczeg József, Ioan Laurentiu Ploscaru, Janzer Olivér, Omer Cerrahoglu, Szabó 789 Barnabás, Szabó 928 Attila, Williams Kada.

• Problems in Mathematics of KöMaL, September 2012

•  Támogatóink: Morgan Stanley