A1. What is the maximum number of odd sums among $x+y, x+z, x+w, y+z, y+w$ and $z+w$, if $x, y, z$ and $w$ are natural numbers?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
A1. We observe that the given sums are precisely all possible sums of 2 out of the numbers $x, y, z$ and $w$. If none of the numbers $x, y, z$ and $w$ is odd, then none of the sums is odd. If exactly 1 of the numbers is odd, then exactly 3 of the sums are odd. If 2 of the numbers are odd, then 4 of the sums are odd. If 3 of the numbers are odd, then exactly 3 of the sums are odd. If all 4 numbers are odd, then no sum is odd. Therefore, the maximum number of odd sums is 4.