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China 2010 quiz2 problem 3

Source:

September 8, 2010

combinatorics unsolvedcombinatorics

Problem Statement

Let

A

be a finite set, and

A_1,A_2,\cdots, A_n

are subsets of

A

with the following conditions: (1)

|A_1|=|A_2|=\cdots=|A_n|=k

, and

k>\frac{|A|}{2}

; (2) for any

a,b\in A

, there exist

A_r,A_s,A_t\,(1\leq r<s<t\leq n)

such that

a,b\in A_r\cap A_s\cap A_t

; (3) for any integer

i,j\, (1\leq i<j\leq n)

,

|A_i\cap A_j|\leq 3

. Find all possible value(s) of

n

when

k

attains maximum among all possible systems

(A_1,A_2,\cdots, A_n,A)

.

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