Question 48
(NET 3) Let $x_1,x_2,x_3,x_4$, and $x_5$ be positive integers satisfying
$$\begin{aligned}
x_1+x_2+x_3+x_4+x_5&=1000,\\
x_1-x_2+x_3-x_4+x_5&\gt 0,\\
x_1+x_2-x_3+x_4-x_5&\gt 0,\\
-x_1+x_2+x_3-x_4+x_5&\gt 0,\\
x_1-x_2+x_3+x_4-x_5&\gt 0,\\
-x_1+x_2-x_3+x_4+x_5&\gt 0.
\end{aligned}$$
(a) Find the maximum of $(x_1+x_3)^{x_2+x_4}$.
(b) In how many different ways can we choose $x_1,\ldots,x_5$ to obtain the desired maximum?
$$\begin{aligned}
x_1+x_2+x_3+x_4+x_5&=1000,\\
x_1-x_2+x_3-x_4+x_5&\gt 0,\\
x_1+x_2-x_3+x_4-x_5&\gt 0,\\
-x_1+x_2+x_3-x_4+x_5&\gt 0,\\
x_1-x_2+x_3+x_4-x_5&\gt 0,\\
-x_1+x_2-x_3+x_4+x_5&\gt 0.
\end{aligned}$$
(a) Find the maximum of $(x_1+x_3)^{x_2+x_4}$.
(b) In how many different ways can we choose $x_1,\ldots,x_5$ to obtain the desired maximum?