MathDB
Good numbers

Source: IberoAmerican Olympiad For University Students

July 10, 2011
linear algebramatrixinductioncombinatorics proposedcombinatorics

Problem Statement

Consider an infinite sequence a1,a2,⋯a_1,a_2,\cdots whose terms all belong to {1,2}\left\{1,2\right\}. A positive integer with nn digits is said to be good if its decimal representation has the form arar+1⋯ar+(n−1)a_ra_{r+1}\cdots a_{r+(n-1)}, for some positive integer rr. Suppose that there are at least 20082008 good numbers with a million digits. Prove that there are at least 20082008 good numbers with 20072007 digits.
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