5

Part of 2000 Putnam

Problems(2)

Source:

Three distinct points with integer coordinates lie in the plane on a circle of radius

r>0

. Show that two of these points are separated by a distance of at least

r^{1/3}

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Source:

S_0

be a finite set of positive integers. We define finite sets

S_1, S_2, \cdots

of positive integers as follows: the integer

a

S_{n+1}

if and only if exactly one of

a-1

a

S_n

. Show that there exist infinitely many integers

N

S_N = S_0 \cup \{ N + a: a \in S_0 \}

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