Problem
COM-B2-M01-P006 An Element in Many Sets
#6
★★★☆☆ Level 3 of 5
There are \(18\) subsets of a \(10\)-element set, each having at least \(4\) elements. Prove that some element belongs to at least \(8\) subsets.
Count incidences \((x,A)\).
The number of pairs \((x,A)\), where \(x\in A\), is at least \(18\cdot4=72\). These pairs are distributed among \(10\) elements. The average number of sets containing an element is at least \(7.2\), so some element belongs to at least \(8\) sets.
Simple, but already an olympiad-style use of averages.