2006 Romanian NMO Problems/Grade 8/Problem 4
Let . Prove that
selected by Mircea Lascu
It is easy to see that the function is convex in each of the three variables (since each term is linear or of the form for each variable ). Thus, its value is maximized at the endpoints. Checking the values of for all possible values of such that yields a maximum of as desired.
As for the minimum, we have
Applying AM-HM to the right hand side yields
Obviously, is maximized when are minimized. That is, when . Thus, we have that