Suppose you have numbers which have the following property : whenever you remove one of them (any one), you can split the remaining ones into two sets having equal sum.
Show that all of these numbers are zero.
a weaker property: whenever you remove one of them (any one), either the sum of the remaining ones is zero or you can split the remaining ones into two sets having equal sum.
this problem is equivalen to this claim that the matrix with is invertible. to prove this claim we'll show that :
where because we're given that so is the set of derangements of
but we know that the number of derangements of a set with even number of elements is odd. so is a sum of odd number of terms where each term is clearly this sum can
never be zero and hence is invertible. Q.E.D.