MA017 2010 HOMEWORK 1 Due 2010-10-05 Tue. (1) Let a 1 ,...,a n be real numbers, and deﬁne f ( x ) = ∑ n j =1 a j sin( jx ). Suppose | f ( x ) | ≤ | sin x | for all real x . Prove, by induction , that | ∑ n j =1 ja j | ≤ 1. (Considering f0 (0) gives a “slick” proof, but I am asking you to use induction.) (2) Show that for any set of ten positive integers less than one hundred, there are two disjoint nonempty subsets having the same sum. (3) Let f0 ( x ) = 1 / (1-x ), and deﬁne f n +1 ( x ) = xf0 n ( x ). Prove that f n ( x ) > 0 for 0 < x < 1 and all n ≥ 0. (4) Given a set of 70 distinct positive integers not exceeding 200, show that two of them diﬀer by 4, 5 or 9. (5) Given any positive integer k ≥ 2, show that we can ﬁnd a sequence of integers
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