Olympiad problems

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1970 Putnam, B3

A closed subset $S$ of $\mathbb{R}^{2}$ lies in $a<x<b$. Show that its projection on the $y$-axis is closed.

Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$, define $S(A)$ to be the set of points that lie on closed segments joining pairs of points of $A$ (a one-point set should be considered to be a special case of a closed segment). For a given nonempty set $A_0$, define $A_n =S(A_{n-1})$ for $n=1,2,\ldots$ Prove that $A_2 =A_3 =\ldots.$

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1970 Putnam, B3

1963 Putnam, B6