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Sum of the digits of x greater than that of 2x

Source: Cono Sur 2004 #3

November 18, 2015
number theorycono sur

Problem Statement

Let nn be a positive integer. We call CnC_n the number of positive integers xx less than 10n10^n such that the sum of the digits of 2x2x is less than the sum of the digits of xx. Show that Cn≥49(10n−1)C_n\geq\frac{4}{9}(10^{n}-1).
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