Author: A.K. Tolpygo
An irrational number \(\alpha\), where \(0 <\alpha <\frac 12\), is given. It defines a new number \(\alpha_1\) as the smaller of the two numbers \(2\alpha\) and \(1 - 2\alpha\). For this number, \(\alpha_2\) is determined similarly, and so on.
a) Prove that for some \(n\) the inequality \(\alpha_n <3/16\) holds.
b) Can it be that \(\alpha_n> 7/40\) for all positive integers \(n\)?
We have a very large chessboard, consisting of white and black squares. We would like to place a stain of a specific shape on this chessboard and we know that the area of this stain is less than the area of one square of the chessboard. Show that it is always possible to place the stain in such a way that it does not cover a vertex of any square.