2006 Romanian NMO Problems/Grade 9/Problem 1
Problem
Find the maximal value of
![$\left( x^3+1 \right) \left( y^3 + 1\right)$](http://latex.artofproblemsolving.com/5/c/9/5c9771df93a5ab9e4e2c14fbf59265a888c74397.png)
where ,
.
Dan Schwarz
Solution
If y is negative, then is also negative, so we want
.
where . Let's see what happens when a gets large:
As a gets large, the fraction gets small, therefore maximizing . But when a gets small(up to 2), the fraction gets bigger, and therefore lessens
.
Therefore, the maximum value of is when x=1 and y=0, which is 2.