Školjka

%V0 $(NET 3)$ Let $x_1, x_2, x_3, x_4,$ and $x_5$ be positive integers satisfying $$x_1 +x_2 +x_3 +x_4 +x_5 = 1000,$$ $$x_1 -x_2 +x_3 -x_4 +x_5 > 0,$$ $$x_1 +x_2 -x_3 +x_4 -x_5 > 0,$$ $$-x_1 +x_2 +x_3 -x_4 +x_5 > 0,$$ $$x_1 -x_2 +x_3 +x_4 -x_5 > 0,$$ $$-x_1 +x_2 -x_3 +x_4 +x_5 > 0$$ $(a)$ Find the maximum of $(x_1 + x_3)^{x_2+x_4}$ $(b)$ In how many different ways can we choose $x_1, . . . , x_5$ to obtain the desired maximum?