MathDB

4

Part of 2018 Romania Team Selection Tests

Problems(3)

(n+k)^2+1 divides factorial

Source: Romanian 2018 TST Day 1 Problem 4

5/25/2020
Given an non-negative integer kk, show that there are infinitely many positive integers nn such that the product of any nn consecutive integers is divisible by (n+k)2+1(n+k)^2+1.
number theorypell equationSophie Germain identityfactorial
Bijection from Z to dZ

Source: Romanian TST Problem 4 Day 2

5/25/2020
Let DD be a non-empty subset of positive integers and let dd be the greatest common divisor of DD, and let dZ=[dn:n∈Z]d\mathbb{Z}=[dn: n \in \mathbb{Z} ]. Prove that there exists a bijection f:Z→dZf: \mathbb{Z} \rightarrow d\mathbb{Z} such that ∣f(n+1)−f(n)∣| f(n+1)-f(n)| is member of DD for every integer nn.
number theorygreatest common divisorbezout s identitynumber theory unsolved
v2 of sum of divisors

Source: Romania 2018 TST Problem 4 Day 3

5/25/2020
Given two positives integers mm and nn, prove that there exists a positive integer kk and a set SS of at least mm multiples of nn such that the numbers 2kσ(s)s\frac {2^k{\sigma({s})}} {s} are odd for every s∈Ss \in S. σ(s)\sigma({s}) is the sum of all positive integers of ss (1 and ss included).
number theorysum of divisors