MathDB

Problem 1

Part of 2010 VJIMC

Problems(2)

convergence sum(1,∞)1/nf(n) for f:N->N

Source: VJIMC 2010 1.1

6/3/2021
a) Is it true that for every bijection f:N→Nf:\mathbb N\to\mathbb N the series ∑n=1∞1nf(n)\sum_{n=1}^\infty\frac1{nf(n)}is convergent? b) Prove that there exists a bijection f:N→Nf:\mathbb N\to\mathbb N such that the series ∑n=1∞1n+f(n)\sum_{n=1}^\infty\frac1{n+f(n)}is convergent. (N\mathbb N is the set of all positive integers.)
SummationConvergence
n=ax+by for given a,b

Source: VJIMC 2010 2.1

6/4/2021
Let aa and bb be given positive coprime integers. Then for every integer nn there exist integers x,yx,y such that n=ax+by.n=ax+by.Prove that n=abn=ab is the greatest integer for which xy≤0xy\le0 in all such representations of nn.
number theory