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Last 2k digits

Source: Iberoamerican Olympiad 2016-P6

September 28, 2016

Problem Statement

Let kk be a positive integer and a1,a2,a_1, a_2, ⋅⋅⋅\cdot \cdot \cdot ,ak, a_k digits. Prove that there exists a positive integer nn such that the last 2k2k digits of 2n2^n are, in the following order, a1,a2,a_1, a_2, ⋅⋅⋅\cdot \cdot \cdot ,ak,b1,b2,, a_k , b_1, b_2, ⋅⋅⋅\cdot \cdot \cdot ,bk, b_k, for certain digits b1,b2,b_1, b_2, ⋅⋅⋅\cdot \cdot \cdot ,bk, b_k
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