One hundred and one numbers are chosen from

| August 30, 2017

Question
1. One hundred and one numbers are chosen from [200] = {1,2,…,200}. Show that one must be a multiple of another. Hint: First show that any natural number can be written in the form (2^k)*a, where k and a are integers such that k>=0 and a>=1 and a is odd.

2. Prove that every set of 12 distinct integers contains a pair whose sum or difference is a multiple of 20.

3. Consider any seven points in a 3 x 4 rectangle. Prove that some pair of points must be separated by a distance less than or equal to sqrt(5).

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