Problem Solving Set 15

18 July 2012

1. Let p be a polynomial with integer coefficients and let a_{1} < a_{2} < ... < a_{k} be integers.

(a) Prove that there exists a ∈ ℤ such that p(a_{i}) divides p(a) for all i = 1, 2, …, k.

(b) Does there exist an a ∈ ℤ such that the product p(a_{1})p(a_{2}) ⋯ p(a_{k}) divides p(a)?

2. S is a set of 15 integers, with the property that for every s ∈ S, there exists a, b ∈ S such that s = a + b. Prove that there exists a non-empty subset T ⊂ S containing 7 or fewer elements whose sum is zero.