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Iterative process on a pair of real numbers

Source: Indian Team Selection Test 2015 Day 4 Problem 2

July 11, 2015
algebra

Problem Statement

Let AA be a finite set of pairs of real numbers such that for any pairs (a,b)(a,b) in AA we have a>0a>0. Let X0=(x0,y0)X_0=(x_0, y_0) be a pair of real numbers(not necessarily from AA). We define Xj+1=(xj+1,yj+1)X_{j+1}=(x_{j+1}, y_{j+1}) for all j≥0j\ge 0 as follows: for all (a,b)∈A(a,b)\in A, if axj+byj>0ax_j+by_j>0 we let Xj+1=XjX_{j+1}=X_j; otherwise we choose a pair (a,b)(a,b) in AA for which axj+byj≤0ax_j+by_j\le 0 and set Xj+1=(xj+a,yj+b)X_{j+1}=(x_j+a, y_j+b). Show that there exists an integer N≥0N\ge 0 such that XN+1=XNX_{N+1}=X_N.
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