MathDB

4

Part of 2015 Tuymaada Olympiad

Problems(2)

Factorial and square

Source: Tuymaada 2015, Day 1, Problem 4, Senior League

7/10/2017
Let n!=ab2n!=ab^2 where aa is free from squares. Prove, that for every ϵ>0\epsilon>0 for every big enough nn it is true, that 2(1−ϵ)n<a<2(1+ϵ)n2^{(1-\epsilon)n}<a<2^{(1+\epsilon)n}
M. Ivanov
factorialalgebra
Decimal representation

Source: Tuymaada 2015, Day 1, Problem 4, Junior League

7/10/2017
Prove that there exists a positive integer nn such that in the decimal representation of each of the numbers n\sqrt{n}, n3,...,n10\sqrt[3]{n},..., \sqrt[10]{n} digits 20152015 stand immediately after the decimal point.
A.Golovanov
number theory