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Does there exists sequence satisfying (mod 10) relation?

Source: 2019 CSMO Grade 11 Problem 1

July 30, 2019
algebra

Problem Statement

Let [a][a] represent the largest integer less than or equal to aa, for any real number aa. Let {a}=a−[a]\{a\} = a - [a].
Are there positive integers m,nm,n and n+1n+1 real numbers x_0,x_1,\hdots,x_n such that x0=428x_0=428, xn=1928x_n=1928, xk+110=[xk10]+m+{xk5}\frac{x_{k+1}}{10} = \left[\frac{x_k}{10}\right] + m + \left\{\frac{x_k}{5}\right\} holds?
Justify your answer.
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