$\displaystyle \\ \mbox{Give that,} \\ x_A , x_B >1 \\ \mbox{ and } \\ Y_A,Y_B\geqslant0 \mbox{ (but strictly positive for at least one)}$

$\displaystyle Y_A + Y_B = 100$

$\displaystyle P_A=Y_A(x_A -1) - Y_B$

$\displaystyle P_B=Y_B(x_B -1) - Y_A$

Problem:

$\displaystyle x_A \mbox{ and } x_B \mbox{ is given } $

$\displaystyle \mbox{Find values of }Y_A, Y_B \mbox{ such that:}$

1)$\displaystyle P_A$ and $\displaystyle P_B$ is always greater than zero. Are there any restictions needed on $\displaystyle x$ for this to be true?Both

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EDIT (21 OCT 07):

Let me clarify what my actual question is. Sorry for not making this clear enough early on.

The question I am asking is this:

1) What restrictions (i.e. the general rule) are needed on $\displaystyle x_A $ and $\displaystyle x_B$ so that there are some range of values of $\displaystyle Y_A$ and $\displaystyle Y_B$ to make both $\displaystyle P_A$ and $\displaystyle P_B$ positive?

2) What is the restriction (i.e. the general rule) for $\displaystyle x_A$ and $\displaystyle x_B$ such that no matter what combination of $\displaystyle Y_A$ and $\displaystyle Y_B$ is chosen (but still sums to 100), $\displaystyle P_A$ and $\displaystyle P_B$ is always negative?

3) How can you prove your answers to questions 1) and 2) ?