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3

Part of 2005 Mediterranean Mathematics Olympiad

Problems(1)

Mediterranian mathematics competition 2005, problem 3

Source: Mediterranian Mathematics Competition 2005, Problem 3

1/8/2006
Let A1,A2,…,AnA_1,A_2,\ldots , A_n (n≥3)(n\geq 3) be finite sets of positive integers. Prove, that 1n(∑i=1n∣Ai∣)+1(n3)∑1≤i<j<k≤n∣Ai∩Aj∩Ak∣≥2(n2)∑1≤i<j≤n∣Ai∩Aj∣ \displaystyle \frac{1}{n} \left( \sum_{i=1}^n |A_i|\right) + \frac{1}{{{n}\choose{3}}}\sum_{1\leq i < j < k \leq n} |A_i \cap A_j \cap A_k| \geq \frac{2}{{{n}\choose{2}}}\sum_{1\leq i < j \leq n}|A_i \cap A_j| holds, where ∣E∣|E| is the cardinality of the set EE
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