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Problem of a Set

Source: 1993 National High School Mathematics League, Exam Two, Problem 2

February 28, 2020

Problem Statement

Set

|A|=n

A_1,A_2,\cdots,A_m

are subsets of

A

, and

A_i\not\subseteq A_j

for any

1\leq i<j\leq m

. Prove: (a)

\sum_{i=1}^{m}\frac{1}{\text{C}_n^{|A_i|}}\leq1

. (b)

\sum_{i=1}^{m}\text{C}_n^{|A_i|}\geq m^2