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Ten pairwise distinct non-zero numbers are such that for each two of them either the sum of these numbers or their product is a rational number.

Prove that the squares of all numbers are rational.

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 \(n\)?