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 .

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 .

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

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

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.

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