I wonder how I can determine the minimum of the product between variables $x$ and $y$ (in terms of $\theta$), given that both $x < 1 - \theta$ and $y < 1 - \theta$, and $x + y = 1$?
So far I can establish that both $\theta < x < 1 - \theta$ and $\theta < y < 1 - \theta$, and that $\theta < 1/2$. What can I say about $xy$ from these?