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A sequence of integers has infinitely many even numbers

Source: INMO 2005 Problem 5

August 23, 2005
floor functionnumber theory unsolvednumber theory

Problem Statement

Let x1x_1 be a given positive integer. A sequence {xn}n≥1\{x_n\}_ {n\geq 1} of positive integers is such that xnx_n, for n≥2n \geq 2, is obtained from xn−1x_ {n-1} by adding some nonzero digit of xn−1x_ {n-1}. Prove that a) the sequence contains an even term; b) the sequence contains infinitely many even terms.
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