MathDB

4

Part of 2018 Benelux

Problems(1)

Benelux Olympiad 2018, Problem 4

Source: BxMO 2018, Problem 4

4/28/2018
An integer n≥2n\geq 2 having exactly ss positive divisors 1=d1<d2<⋯<ds=n1=d_1<d_2<\cdots<d_s=n is said to be good if there exists an integer kk, with 2≤k≤s2\leq k\leq s, such that dk>1+d1+⋯+dk−1d_k>1+d_1+\cdots+d_{k-1}. An integer n≥2n\geq 2 is said to be bad if it is not good. (a) Show that there are infinitely many bad integers. (b) Prove that, among any seven consecutive integers all greater than 22, there are always at least four good integers. (c) Show that there are infinitely many sequences of seven consecutive good integers.
BxMOBeneluxnumber theory