2006 Romanian NMO Problems/Grade 9/Problem 1


Find the maximal value of

$\left( x^3+1 \right) \left( y^3 + 1\right)$,

where $x,y \in \mathbb R$, $x+y=1$.

Dan Schwarz


If y is negative, then $(x^3+1)(y^3+1)$ is also negative, so we want $0\leq y\leq 1$.


where $1 \leq a$. Let's see what happens when a gets large:



As a gets large, the fraction gets small, therefore maximizing $(x^3+1)(y^3+1)$. But when a gets small(up to 2), the fraction gets bigger, and therefore lessens $(x^3+1)(y^3+1)$.

Therefore, the maximum value of $(x^3+1)(y^3+1)$ is when x=1 and y=0, which is 2.

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