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Putnam 1977 B3

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April 7, 2022
college contests

Problem Statement

An (ordered) triple (x1,x2,x3)(x_1,x_2,x_3) of positive irrational numbers with x1+x2+x3=1x_1+x_2+x_3=1 is called balanced if each xi<1/2.x_i< 1/2. If a triple is not balanced, say if xj>1/2x_j>1/2, one performs the following balancing act B(x1,x2,x3)=(x1′,x2′,x3′),B(x_1,x_2,x_3)=(x'_1,x'_2,x'_3), where xi′=2xix'_i=2x_i if i≠ji\neq j and xj′=2xj−1.x'_j=2x_j-1. If the new triple is not balanced, one performs the balancing act on it. Does the continuation of this process always lead to a balanced triple after a finite number of performances of the balancing act?
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