The American Mathematical Monthly, Vol. 99, No. 3 (1992), pp. 256-258.

Ömer Egecioglu

A Combinatorial Generalization of a Putnam Problem

Let $ a_1 , a_2 , \ldots , a_{2n+1} $ be a sequence of integers such that, if any of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_1 = a_2 = \cdots = a_{2n+1}$

Here we give a combinatorial proof of a generalization of this problem. The arguments rely on a matrix theoretic formulation of the original problem and elementary properties of cyclotomic polynomials.