My solution of POW12 was accepted. [Remark of December 19, 2009: it is only stated that it was “completely or partially proved”]
“ Problem. Find, with proof, the maximum value of where and is variable. In particular, your answer must be greater than or equal to all values obtained from other choices of “
Here is the solution I submited.
Let , , and . For a given , with , we know by the Lagrange multipliers method that is a local extremum if for
where is the value of the multiplier that is a solution of these equations. Hence we have respectively
Solving this system of equations we find
the latter being a local extremum of , for a fixed . We transformed the initial maximizing problem in continuous variables and the discrete variable into the maximizing of . Now we introduce the following function
The function has a maximum at the same point than the function
On the other hand for for and for . Therefore
is a maximum of . Since , the maximum occurs at . Thus, for we have
Update (Dec., 19,2009): some errors (identified by Rod Carvalho here) corrected.