Homework 7

Deadline: Dec. 14, 22:00:00
  1. Prove the even/odd town theorem:

    Suppose there are clubs and people such that (i) each club contains an even number of people and (ii) each pair of clubs shares an odd number of people. Then .

  2. Prove the odd/odd town theorem:

    Suppose there are clubs and people such that (i) each club contains an odd number of people and (ii) each pair of clubs shares an odd number of people. Then .

  3. Suppose is a family of set of such that (i) , is not divisible by , and (ii) , is divisible by . Show that .

  4. Suppose are two sets of points in such that all distances between members of and members of are equal. Prove that .

  5. We have balls of some weights. Suppose that if we remove any of them, the remaining weights can be divided into two sets of , with equal weight. Prove that all balls have the same weight.

    (Hint: Suppose is an matrix with zeros in the diagonal and everywhere else. Then has an inverse matrix. (Prove this result before using it.))