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MMO 364 Moscow MO 1957 no of digits 1,2,3,...,10&8 equals 0s in 1,2,3, ...,10^9

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March 20, 2021
algebranumber theory

Problem Statement

(a) Prove that the number of all digits in the sequence 1,2,3,...,1081, 2, 3,... , 10^8 is equal to the number of all zeros in the sequence 1,2,3,...,1091, 2, 3, ... , 10^9.
(b) Prove that the number of all digits in the sequence 1,2,3,...,10k1, 2, 3, ... , 10^k is equal to the number of all zeros in the sequence 1,2,3,...,10k+11, 2, 3, ... , 10^{k+1}.
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