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Interesting process

Source: Serbia TST 2017 #2

May 21, 2017
combinatorics

Problem Statement

Initally a pair (x,y)(x, y) is written on the board, such that exactly one of it's coordinates is odd. On such a pair we perform an operation to get pair (x2,y+x2)(\frac x 2, y+\frac x 2) if 2∣x2|x and (x+y2,y2)(x+\frac y 2, \frac y 2) if 2∣y2|y. Prove that for every odd n>1n>1 there is a even positive integer b<nb<n such that starting from the pair (n,b)(n, b) we will get the pair (b,n)(b, n) after finitely many operations.
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