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 .
Deadline expired on October 10, 2012.
Solution. (a) For k3, apply the AM-GM inequality to 1/(k-2) and ak/2 with the weights k-2 and 2:
We have equality if 1/(k-2)=ak/2, so .
Taking the product for ,
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
7 students sent a solution. 5 points: Herczeg József, Ioan Laurentiu Ploscaru, Janzer Olivér, Omer Cerrahoglu, Szabó 789 Barnabás, Szabó 928 Attila, Williams Kada.