MathDB

Problem 4

Part of 2010 VJIMC

Problems(2)

weaker abundance of a number

Source: VJIMC 2010 1.4

6/4/2021
For every positive integer nn let σ(n)\sigma(n) denote the sum of all its positive divisors. A number nn is called weird if σ(n)≥2n\sigma(n)\ge2n and there exists no representation n=d1+d2+…+dr,n=d_1+d_2+\ldots+d_r,where r>1r>1 and d1,…,drd_1,\ldots,d_r are pairwise distinct positive divisors of nn. Prove that there are infinitely many weird numbers.
number theory
image as a subset of rectangles

Source: VJIMC 2010 2.4

6/4/2021
Let f:[0,1]→Rf:[0,1]\to\mathbb R be a function satisfying ∣f(x)−f(y)∣≤∣x−y∣|f(x)-f(y)|\le|x-y|for every x,y∈[0,1]x,y\in[0,1]. Show that for every ε>0\varepsilon>0 there exists a countable family of rectangles (Ri)(R_i) of dimensions ai×bia_i\times b_i, ai≤bia_i\le b_i in the plane such that {(x,f(x)):x∈[0,1]}⊂⋃iRi and ∑iai<ε.\{(x,f(x)):x\in[0,1]\}\subset\bigcup_iR_i\text{ and }\sum_ia_i<\varepsilon.(The edges of the rectangles are not necessarily parallel to the coordinate axes.)
functiongeometryrectangle