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ThePerfectHacker · #1 June 4th, 2007, 01:11 PM

1)Let $S_1$ be a square in the coordinate plane. Divide this square into 4 equal squares by drawing lines straight down the middle. Pick any one of the smaller squares, call it $S_2$ . Now divide this square into 4 smaller squares, pick any one, call it $S_3$ . And thus on. Let $s_1,s_2,s_3,...$ be the sequence of points which represent the centers of $S_1,S_2,S_3,...$ respectively. Show that $(s_n)$ convergences to some point.

2)Let $U$ be a subset of $\mathbb{R}$ which is closed under multiplication*. Let $S \mbox{ and }T$ two disjoint sets whose union is $U$ . With the property that the product of any three elements is again in the set. Show that one of the sets $S,T$ must be closed under multiplication.

3)Let $x$ be a non-zero real number so that $x+x^{-1}$ is an integer. Show that $x^n + x^{-n}$ is an integer for every integer $n$ .

*)Meaning, if $a,b\in U$ then $ab\in U$ .