Problem 2

Let

n

be positive and let

A_1,\ldots,A_{2n+1}

be subsets of

B

. Suppose each

A_i

has exactly

2n

elements, every two distinct

A_i

have exactly one common element, and every element of

B

belongs to at least two

A_i

. For which

n

can one label each element 0 or 1 so that each

A_i

has 0 on exactly

n

elements?

Step 1 of 5: Step 1

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A_i\text{ meets each other }A_j\text{ once}

Detailed analysis

Each element of A_i belongs to another set. A fixed other set contains at most one element of A_i, and there are 2n other sets; since A_i has 2n elements, each other set contains exactly one. Thus each element belongs to exactly two sets.