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Problem 1

Part of 1970 Yugoslav Team Selection Test

Problems(1)

NT inequality, number of digits

Source: Yugoslav TST 1970 P1

5/30/2021
Positive integers aa and bb have nn digits each in their decimal representation. Assume that mm is a positive integer such that n2<m<n\frac n2<m<n and assume that each of the leftmost mm digits of aa is equal to the corresponding digit of bb. Prove that a1n−b1n<1n.a^{\frac1n}-b^{\frac1n}<\frac1n.
inequalitiesnumber theory