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Colouring digits to make a rational Number

Source: India EGMO 2022 TST P4

November 28, 2021
number theorycombinatorics

Problem Statement

Let NN be a positive integer. Suppose given any real x∈(0,1)x\in (0,1) with decimal representation 0.a1a2a3a4⋯0.a_1a_2a_3a_4\cdots, one can color the digits a1,a2,⋯a_1,a_2,\cdots with NN colors so that the following hold: 1. each color is used at least once; 2. for any color, if we delete all the digits in xx except those of this color, the resulting decimal number is rational. Find the least possible value of NN.
~Sutanay Bhattacharya
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