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5

Part of 1998 Putnam

Problems(2)

1998 Putnam A5

Source:

4/19/2013
Let F\mathcal{F} be a finite collection of open discs in R2\mathbb{R}^2 whose union contains a set E⊆R2E\subseteq \mathbb{R}^2. Show that there is a pairwise disjoint subcollection D1,…,DnD_1,\ldots,D_n in F\mathcal{F} such that E⊆∪j=1n3Dj.E\subseteq\cup_{j=1}^n 3D_j. Here, if DD is the disc of radius rr and center PP, then 3D3D is the disc of radius 3r3r and center PP.
Putnam 1998 B5

Source:

4/7/2012
Let NN be the positive integer with 1998 decimal digits, all of them 1; that is, N=1111⋯11.N=1111\cdots 11. Find the thousandth digit after the decimal point of N\sqrt N.
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