Suppose > 0 and k > 1. We show that if n > n₀(k, ) and A Z_n satisfies A > ((1/k) + )n then there is a subset B A such that 0 < B ≤ k and Σ_bBb = 0 (in Z_n). The case k = 3 solves a problem of Stalley and another problem of Erdös and Graham. For an integer m > 0, let snd(m) denote the smallest integer that does not divide m. We prove that for every > 0 there is a constant c = c() > 1, such that for every n > 0 and every m, n^{1 +} ≤ m ≤ n²/log²n every set A {1, 2,…, n} of cardinality A >c · n/snd(m) contains a subset B A so that Σ_bBb = m. This is best possible, up to the constant c. In particular it implies that for every n there is an m such that every set A {1,…,n} of cardinality A >cn/log n contains a subset B A so that Σ_bBb = m, thus settling a problem of Erdös and Graham.