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Let S be a set of rational numbers with the following properties:
1) 1/2 is an element of S
2) If x is an element of S, then both 1/(x+1) is an element of S and x/(x+1) is
an element of S
Prove that S contains all rational numbers in the interval 0<x<1.
Source: unknown
Cut the knot: Euclid's Algorithm.
The Broken Calculator Puzzle. In this puzzle, a calculator initially showing 0 can be made to show any rational number by pressing sin, cos, tan and their inverses in some order a finite number of times. The method and the proof are similar to this puzzle.
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